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# Wildlife Population Dynamics FW 662

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FW662 Lecture 13 Management Lecture 13 Management of populations Reading Hilborn R and C J Walters Chapter 3 Behavior of exploited populations Pages 47 103 in Quantitative sheries stock assessment Chapman and Hall New York New York USA 570 pp Ludwig D R Hilborn and C Walters 1993 Uncertainty resource exploitation and conservation lessons from history Science 2601736 Stacey P B and M Taper 1992 Environmental variation and the persistence of small populations Ecological Applications 2 1829 Optional Anderson D R 1985 Constrained optimal exploitation a quantitative theory Pages 105116 in S L Beasom and S F Roberson Game Harvest Management McCullough D R 1984 Lessons from the George Reserve Michigan Pages 211242 in Whitetailed deer ecology and management Wildlife Management Institute Walter C 1986 Chapter 4 Models of Renewable Resource Systems Pages 64128 in Adaptive Management of Renewable Resources Macmillian New York New York USA 374 pp Management of a population requires 4 steps Hilborn and Walters 1992 de nition of a goal development of a model to evaluate management options to achieve the goal implementation of the management option selected along with necessary data collection schemes and an evaluation procedure to see that the management strategy is working 1 Goal or objective some objectives may be hidden Examples are Threatened and endangered species raise the population level to insure persistence Minimum viable population Persistence Time to extinction Pest control lower the population or more reasonably lower the level of damage same number of coyotes only the ones present don t like mutton Commercially important species such as halibut are managed for maximum sustained yield Game species Maximize production of trophy animals Maximize hunter recreation generating maximum income Maximize quality of recreation which may increase license cost Minimize game damage payments andor rancher complaints 2 Need a model to be able to develop your management strategies Once you have de ned a goal you have to have some ability to test various management strategies decisions to see what level will achieve the goal Anderson 1985 concerning exploitation argues that to develop an optimal FW662 Lecture 13 Management 2 exploitation system you need a birth process as a function of density and b death process as a function of density and number harvested a and b constitute densitydependent relationships in the population He also advocates including the environmental stochasticity inherent in the natural system in the model He argues that deterministic models or simple models such as the logistic where r b d are totally inadequate He provides the understatement of the century in population biology quotThe availability of adequate data will continue to be a serious limitationquot In reality lack of information is deeper than just some poorly estimated parameters Often basic functional relationships are poorly understood and are not modeled adequately to provide correct system behavior regardless of the parameter values used Stacey and Taper 1992 concerning population persistence also argue for models with a environmental stochasticity and b density dependence including the proper form of the density dependence In persistence models as population declines the compensation for small population size takes the form of increased birth rates and decreased death rates density dependence and so is a significant factor in increasing population persistence Stacey and Taper 1992 tested 2 forms of density dependence with their data logistic 7 w Rt 7111 K Rt RI 1 w e K and Blogistic However their data precluded a significant test between these models Their data did show significant correlations between adult survival and population size although the lack of correlation is likely a Type II error and additional analyses are provided by Middleton and Nisbet 1997 In the following table 4 variables are correlated against population size Variable Sample Correlation Probability Size Adult Survival 9 065 0058 Juvenile Surv 9 030 0434 Reproduction 10 0 56 0094 FW662 Lecture 13 Management 3 Emigration 9 028 0473 Stacey and Taper 1992 also make some grandstand statements about data required to build a model quotit can be exceedingly difficult to provide meaningful estimates of persistence time because the results depend so largely on the assumptions that are used to create the modelquot What a revelation However they continue quotFurthermore even if the model is speci ed correctly relatively small errors in the estimation of the parameters can lead to large errors in the predictions see also Goodman 1984quot They explored 2 functions for incorporating density dependence illustrating a lack of knowledge on functional forms of a very basic relationship To summarize the need for a model is to guide management explore options evaluate options quantitatively and to formulate optimal decision criterion Even if the model is built from very poor data it may be useful in guiding collection of the needed data 3 Ready to implement our management scheme a Need annual data to maintain an optimal decision Anderson 1985 Examples are to estimate population size with rigorous estimation schemes or maybe estimate reproductive rates via helicopter surveys such as CDOW does with deer and elk or the USFWS and the Canadian Wildlife Service does with air surveys of pothole density and brood density or maybe estimate annual survival rates via radiocollared animals or bands or estimate harvest taken or maybe immigrationemigration rates to maintain a metapopulation An example of a very simple data collection scheme is McCullough et al39s 1990 linked sex harvest strategy Hunters are able to shoot animals of either sex but antlered males are preferred For a constant population size the proportion of males in the harvest is a function of harvest size FW662 Lecture 13 Management 4 41 U I E Linked Sex Harvest 03 EE 2 U l 3939 U 2 4 C39 D O 3 Si 0 2o 40 6O 80 100 Number Harvested Incremental increases in harvest are used to seek out a predetermined percentage of males in the harvest As the harvest size increases the percent of females in the harvest increases When too many females are taken the population eventually declines below MSY and the proportion of females increases even more in the harvest Typically try to adjust the harvest so that 40 of the harvest is female but this value will depend on the herd productivity and recruitment rates The least attractive aspect of this strategy is that no additional monitoring data are taken to verify that the system is operating as expected A stochastic environment could really screw up the works ie a year of really poor overwinter fawn survival In 2 years you would have no bucks and hence see a very large percentage of females in the harvest Unfortunately the strategy turns out to work poorly even in a deterministic model Lubow et al 1996 To understand the problem consider what is needed to reach the population size for MSY If I told you the exact population size and age and sex ratios plus the true values of the parameters K and r and that the population truly followed a logistic curve ie MSY population size is K2 you could in theory immediately reach N MSY This is because we know under this set of assumptions that N MSY K2 Now I could only tell you the proportion of adult females to harvest at MSY This would imply that I didn t tell you K as above but only r because we know that at MSY the harvest rate is r2 ie the MSY harvest is rK4 NMSYrZ You would see a gradual decline in N until NMSY was reached The time lag will be on the order of a generation time or so but not FW662 Lecture 13 Management 5 terribly long However even less information is provided if I were to only tell you the ratio of females to males to harvest at MSY If you implemented this sex ratio in your current population the time lag to achieve the MSY population size is on the order of 10 generation times Hence you would probably want additional information to invoke this harvest strategy like the current population size and K b A second part of the implementation process is obtaining public acceptance There are classic cases where elaborate management plans have failed For instance suppose deer hunters would not shoot does under the linked sex strategy described above Other documented failures are antlerpoint restrictions in deer management where many animals are harvested but left in the field because they are illegal In the spotted owl controversy loggers will not accept the proposed guidelines for maintaining the owl population because they interfere with tree harvest Finally the Colorado black bear controversy Amendment 10 is an example of lack of public acceptance Because spring bear harvest is heavily biased towards the taking of males they come out of hibernation first the spring harvest can result in a larger bear harvest than fall hunting Spring bear hunting over bait was an acceptable practice 20 years ago Now the public has changed it view The killing of a sow with newborn cubs is unacceptable From public surveys the CDOW found out in the spring of 1992 that 40 of Colorado voters strongly opposed spring bear hunting and another 14 opposed spring bear hunting Prior to the amendment being placed on the ballot 69 said they opposed spring bear hunting and would vote to support the amendment banning spring bear hunting Other public surveys showed that 21 agreed with the statement quotHunting is a righ quot 57 agreed with the statement quotLegal hunting is okayquot 15 agreed with the statement quotHunting is okay as long as done by wildlife professionalsquot and 7 agreed with the statement quotNo hunting should be allowed period on any speciesquot The Colorado Wildlife Commission ignored these results and persisted with the spring bear season even though the CDOW biologists strongly recommended against a spring season In the 1992 November election 70 of the public supported Amendment 10 that banned spring bear hunting Other aspects of understanding the public is to understand the dynamics of the human population affected by the management strategy Hilbom and Walters 1992 have a chapter on the dynamics of the fishing eet Reasons why economic considerations may lead to over exploitation of a natural resource from Bulmer 1994 120 taken from Clark 1990 i In a monopoly situation soleowners of a resource stock tend to view it as a capital asset that is expected to earn dividends at the current rate FW662 Lecture 13 Management 6 ii In an openaccess situation in which the resource is the common property of a group of competing users each of these users will consider only his own interests and will fail to take into account the costs that his actions may impose on the other users 4 Periodic evaluation of the management strategy must be performed along with incorporation of new information and design of management experiments to test the validity of the strategy As an example of the negative side of a management strategy consider the following shortcomings of MSY Holt and Talbot 1978 focuses attention on the dynamics of particular species or stocks without explicit regard to the interactions between those species or stocks and other components of the ecosystem concerns only the quantity and not the quality of potential yield or other value from the resource depends on a degree of stability and resilience of the resource that may not exist focuses attention on the output from resource use without regard to the input of energy or other natural resources and of human skill and labor required to secure the output may admit and even encourage over exploitation Should have criteria specified a priori to evaluate the management program This idea along with extending knowledge is particularly well developed by Walters 1986 He suggests 4 basic issues in developing an adaptive management strategy bounding of management problems in terms of explicit and hidden objectives practical constraints on action and the breath of factors considered in policy analysis represent existing understanding of managed systems in terms of more explicit models of dynamic behavior that spell out assumptions and predictions clearly enough so that errors can be detected and used as a basis for further learning representation of uncertainty and its propagation through time in relation to management actions using statistical measures and imaginative identification of alternative hypotheses models that are consistent with experience but might point toward opportunities for improved productivity design of balanced policies that provide for continuing resource production while simultaneously probing for better understanding and untested opportunity FW662 Lecture 13 Management 7 The major contribution is to use management as an experiment to more clearly understand the system being managed Some additional references on adaptive management Haney A and R L Power 1996 Adaptive management for sound ecosystem management Environmental Management 20879886 Holling CS 1978 Adaptive environmental assessment and management John Wiley London United Kingdom Lancia R A et al 1996 ARM For the future adaptive resource management in the wildlife profession Wildlife Society Bulletin 24436442 McLain R J and R G Lee 1996 Adaptive management promises and pitfalls Environmental Management 20437448 Noss R F and A Y Cooperrider 1995 Saving nature39s legacy protecting and restoring biodiversity Island Press see chapter 9 on monitoring Van Winkle W et al 1997 Uncertainty and instream ow standards perspectives based on hydropower research and assessment Fisheries 222122 Examples of population management CDOW big game management Primary objective is to maintain the population at a specified size agreed upon by CDOW habitat agencies USFS BLM and locals A secondary objective is to maximize hunter recreation that has the hidden objective of maximizing revenue Some DAU39s Data Analysis Units are specified licenses meaning that only a limited number of hunters are given licenses to hunt in these units This is an attempt to maximize quality of recreation The remainder of the units are unlimited ie anyone can purchase a license and hunt CDOW can assume that most of the male segment of the herd will be harvested with these overthecounter licenses Only a limited number of licenses for harvest of females are provided with the objective of holding the population at the herd objective Data collected December age and sex ratios providing reproduction rates and the number of males left after harvest January quadrat counts providing population size Both of the above are conducted via helicopter Harvest estimates conducted by phone over the counter licenses and mail limited licenses In March these data are used in the POPII model to determine the harvest needed FW662 Lecture 13 Management 8 in the upcoming season Major weaknesses of the POPII model are that it lacks density dependence because so little is known to specify a reasonable function Environmental stochasticity is speci ed as the quotguessedquot survival rates to obtain alignment on observed data Various harvest levels for the upcoming season are tried to maintain the population at the herd objective for the coming year One of the concerns with game harvest particularly the selective harvest of ungulates with large antlers is the impact on the genetics of the population from continued selection Ratner and Lande 2001 found with a model that selective harvesting based on size can produce evolutionary changes in equilibrium mean size and abundance This should come as no surprise as strong selective pressure is going to cause a change in the population Harris et al 2002 discuss four potential effects from sport hunting 1 it may alter the rate of gene ow among neighboring demes 2 it may alter the rate of genetic drift through its effect on genetically effective population size 3 it may decrease fitness by deliberately culling individuals with traits deemed undesirable by hunters or managers and 4 it may inadvertently decrease fitness by selectively removing individuals with traits desired by hunters Studies specifically investigating these issues have been rare but undesirable genetic consequences from hunting have been documented in only a few cases North American Waterfowl populations Anderson 1 Funding Duck Stamp Program Federal Aid Program 2 Surveys Breeding Ground Conditions 1955 Breeding Population Size 1955 Harvest Estimates 1954 Age amp Sex Ratio Estimates 1961 Size of Wintering Populations 1950 Special Surveys 3 Models critical since about 1969 Production Models Pond Models Population Models nonlogistic 4 Decision Making National and Flyway Waterfowl Councils NGOs Audubon DU USDI and FWS Committees US and CWS and States and Provinces 5 Optimal Management A new initiative Additional Modeling FW662 Lecture 13 Management 9 Alternative Objectives Stochastic Dynamic Programming Conclusions about population management Need good experimental manipulations to achieve models with the quality to be useful in managing populations Nichols 1991 Abstract This essay deals with the relevance of some of the ideas of Romesburg 1981 to population ecology and management of the American black duck Arias rubripes Most investigations dealing with the effects of hunting regulations on black duck populations have used the hypotheticodeductive HD approach of specifying a priori 39 n quot and 39 J deduced J J39 quot These investigations have not used manipulative experimentation however but have involved severely constrained analyses of historical data and have thus produced weak inferences The 1982 lawsuit over black duck hunting regulations the current uncertainty about appropriate black duck management actions and the frequent skirmishes in the published literature of black duck population ecology are natural consequences of these weak inferences I suggest that we attempt to take advantage of management and other manipulations by treating them as an opportunity to learn something via experimentation as recommended by Macnab 1983 and Walters 1986 Sinclair 1991 Abstract This essay explains the need for wildlife management as scientific experiments to achieve reliable knowledge Romesburg 1981 and emphasizes that science and management are not alternative processes I explain the rationale behind the scientific method the construction of hypotheses and their predictions and how to test them with manipulations available through the management of wildlife The scale of wildlife management programs makes them suitable for scientific experimentation Macnab 1983 Problems such as population regulation and predatorprey interactions are used to show that theory is needed to develop proper predictions Reasons for the failure of simple MSY harvesting strategies The MSY harvesting strategy assumes that managers know K the current population size N and can therefore harvest exactly the number of animals to maintain the population at MSY However populations are not deterministic so populations do not grow exactly the same each year ie the process variance is not zero Further managers are usually not able to know the critical parameters of the population ie r and K and are not able to know N each year but only get an estimate A7 As a result following a MSY harvest strategy may lead to unsustainable harvest levels Aanes et al 2002 evaluated alternative harvest strategies for a willow ptarmigan Lagopus lagopus population on a private estate in Sweden They fit a thetalogistic model to 32 years of data FW662 Lecture 13 Management 10 r logNM logN 7 H 17 1 NITHI e K 9 K where r is the rate of increase when there is lbird is in the population and K is carrying capacity They considered 5 harvest strategies 1 constant harvesting where a xed number of birds were removed each year ie H constant 2 proportional harvesting where a constant fraction of birds were removed each year ie H t cNt 3 restricted proportional harvesting where an upper limit is introduced on proportional harvesting 4 threshold harvesting where the estimated number of birds greater than a threshold population is harvested and 5 proportional threshold harvesting where only a xed proportion of the birds above the threshold are harvested ie harvesting only a xed proportion of the difference between the estimated population size and the threshold when this difference is positive Restricted proportional harvesting gave slightly higher mean annual yields than proportional threshold harvesting However variance in annual yield was reduced by restricted proportional harvesting because periods with low population size became shorter Uncertainties in population parameters did not affect which strategy was optimal although those uncertainties strongly in uenced the expected yield and the uncertainties in the hunting statistics Aanes et al 2002 Of course this last conclusion will be a function of the variance of the population estimates and so is somewhat speci c to this study Literature Cited Aanes S S Engen BE Seether T Willebrand and V Marcstrom 2002 Sustainable harvesting strategies of willow ptarmigan in a uctuating environment Ecological Applications 12281290 Anderson D R 1985 Constrained optimal exploitation a quantitative theory Pages 105116 in S L Beasom and S F Roberson Game Harvest Management Bulmer M 1994 Theoretical evolutionary ecology Sinauer Associates Inc Sunderland Massachusetts USA 352 pp Clark C 1990 Mathematical bioeconomics the optimal management of renewable resources 2nd ed Wiley New York New York USA Harris R B W A Wall and F W Allendorf 2002 Genetic consequences of hunting what do we know and what should we do Wildlife Society Bulletin 30634643 Hilbom R and C J Walters 1992 Behavior of exploited populations Pages 47103 in Quantitative sheries stock assessment Chapman and Hall New York New York USA 570 pp FW662 Lecture 13 Management 11 Holt S J and L M Talbot 1978 New principles for the conservation of wild living resources Wildlife Monograph 59 33 pp Lubow B C G C White and D R Anderson 1996 Evaluation ofa linked sex harvest strategy for cervid populations Journal of Wildlife Management 60787796 Ludwig D R Hilbom and C Walters 1993 Uncertainty resource exploitation and conservation lessons from history Science 2601736 Macnab J 1983 Wildlife management as scientific experimentation Wildlife Society Bulletin 1 1397401 McCullough D R 1984 Lessons from the George Reserve Michigan Pages 211242 in Whitetailed deer ecology and management Wildlife Management Institute McCullough D R D S Pine D L Whitmore T M Mansfield and R H Decker 1990 Linked sex harvest strategy for big game management with a test case on blacktailed deer Wildlife Monograph 112 41 pp Middleton D A J and R M Nisbet 1997 Population persistence timeestimates models and mechanisms Ecological Applications 7 107117 Nichols J D 1991 Science population ecology and the management of the American black duck Journal of Wildlife Management 55790 799 Ratner S and R Lande 2001 Demographic and evolutionary responses to selective harvesting in populations with discrete generations Ecology 8230933104 Romesburg H C 1981 Wildlife science gaining reliable knowledge Journal of Wildlife Management 45293313 Sinclair A R E 1991 Science and the practice of wildlife management Journal of Wildlife Management 55767773 Stacey P B and M Taper 1992 Environmental variation and the persistence of small populations Ecological Applications 2 1829 Walter C 1986 Chapter 4 Models of Renewable Resource Systems Pages 64128 in Adaptive Management of Renewable Resources Macmillian New York New York USA 374 pp FW662 Lecture 9 Immigration and Emigration 1 Lecture 9 Role of immigration and emigration in populations Reading Sinclair A R E 1992 Do large mammals disperse like small mammals Pages 229 242 in Stenseth N C and W Z Lidicker Jr Eds Animal dispersal small mammals as a model Chapman and Hall New York New York USA 365 pp Optional Greenwood P J 1983 Chapter 7 Mating systems and the evolutionary consequences of dispersal Pages 116131 in I R Swingland and P J Greenwood eds The ecology of animal movement Clarendon Press Oxford England Stenseth N C 1983 Chapter 5 Causes and consequences of dispersal in small mammals Pages 63101 in I R Swingland and P J Greenwood eds The ecology of animal movement Clarendon Press Oxford England Dispersal waif of population ecology Ricklefs 1990373 Important but difficult to measure Dispersal is defined as the oneway permanent movement away from an established home range or natal area In contrast migration is the twoway movement between 2 areas Philopatry is the fidelity or tenacity to an area or home range Immigration can be thought of as dispersing animals that are leaving the area of interest whereas emigration is the arrival of dispersing animals onto the area of interest Two types of dispersal Lidicker 1975 Saturation dispersal population crowded with respect to resources and aggressive individuals force others to leave Presaturation dispersal individuals with an innate predisposition to wander and leave Dispersers leave their current place of residency before the patch s carrying capacity is reached FW662 Lecture 9 Immigration and Emigration 2 Saturation Dispersal 100 40 60 Time t Proximate causes of dispersal competition for mates avoidance of inbreeding Bollinger et a1 1993 and competition for resources Boolinger et a1 1993 found that meadow voles Microtus permsylvanicus released into experimental grassland plots with siblings were more likely to disperse from these plots than were voles released into similar plots with nonsiblings Furthermore voles that dispersed from sibling groups did so sooner than dispersing voles from nonsibling groups Dispersal how the mechanism evolved see Emlen 1984 Chapter 13 Evolve a genotype that leaves an area where population is close to carrying capacity This genotype would be deleterious at low densities but bene cial at high densities assuming areas away where tness is increased Role in population regulation Keeps population from getting too high in one area Evidence for population regulation provided by quotfence effectquot Krebs 1992 Buffers population over several subpopulations Maintains genetic structure Maximum dispersal takes place at MSY when there are the most juveniles available to disperse at least according to common perceptions However this perception has not held up with experimental work Andreassen and Ims 2001 in an experimental study with 12 enclosed patchy populations found that dispersal in root voles Microtus oeconomus was strongly densitydependent and most so for subadult animals However highdensity patches had low emigration rates Root voles immigrated onto patches with a smaller number of individuals especially of FW662 Lecture 9 Immigration and Emigration 3 their own sex and reproductive state than were present in the patch they left Most shifts between patches took place from patches with relatively low density to patches with even lower density Small patches had higher spatiotemporal variability in density and demographic composition than large patches and this probably caused most of the demographic turnover in small patches In particular emigration was the main demographic parameter behind declining numbers and patch extinction in small patches with few individuals The kind of density dependent emigrationimmigration dynamics found by Andreassen and Ims 2001 does not match the common perception that dispersal works primarily to reduce extinction probabilities through rescue effects In particular the impact of emigration as a factor that may increase the extinction probability of small isolated patches with few individuals is an important aspect of metapopulations Dispersers should be either young animals which have not yet attained maximal reproductive value and which have not yet established a breeding site or much older individuals whose prospects for future reproductive output is low but still greater than zero Morris 1982 Examples of older individuals leaving are male lions from pride or 10 year old African buffalo males leave the breeding herds Sinclair 1992 Applications of dispersal theory in applied ecology Hansson 1992 Generally immigration is more pertinent than emigration Pest control removal of pest and time to recolonization Pest outbreaks iack rabbits house mice Settling in human habitations annual fall mouse trapping episodes Disease transmission both within the species and to other species eg livestock humans Recolonization of areas with extinct populations Yellowstone wolf is it necessary to transplant individuals Dispersal distances how to measure Porter and Dooley 1993 Most studies get a biased view of dispersal distances because of distanceweighted sampling procedures Radios provide the best method for estimating dispersal distances but even new technology can be awed if the animals move far enough Dispersal distances are important in metapopulation models because distance along with frequency determines time to recolonization of a vacant patch and gene ow frequencies Good example of a study with radios to measure dispersal is Larsen and Boutin 1994 on 250 red squirrels Tamiasciurus hudsonicus in Alberta Canada Foray distance was not related to age or size of the offspring Offspring that settled relatively farther away from their natal territory were more likely to obtain larger territories with traditional hoarding and overwintering sites middens These offspring also had higher overwinter survival suggesting that the costs of making forays off the natal territory may be balanced by the advantages of locating a superior territory FW662 Lecture 9 Immigration and Emigration 4 Modeling dispersal and population dynamics Hastings 1993 Dispersal stabilizes chaotic discrete logistic models A study of one of the simplest systems incorporating both dispersal and local dynamics coupling two discrete time logistic equations demonstrates several surprising features Passive dispersal can cause chaotic dynamics to be replaced by simple periodic dynamics Thus passive movement can be stabilizing even in a deterministic model without underlying spatial variation in the dynamics The boundary between initial conditions leading to qualitatively different dynamics can be a fractal so it is essentially impossible to specify the asymptotic behavior in terms of the initial conditions In accord with several recent studies of arthropods and earlier theoretical work density dependence may only be detectable at a small enough spatial scale so efforts to uncover density dependence must include investigations of movemen Again a superficially simple model suggests a hypothesis about real populations that will be difficult to test Literature Cited Andreassen H P and R A Ims 2001 Dispersal in patchy vole populations role of patch configuration density dependence and demography Ecology 8229112926 Bollinger E K S J Harper and G W Barrett 1993 Inbreeding avoidance increases dispersal movements ofthe meadow vole Ecology 74 1 1531 156 Hansson L 1992 Small mammal dispersal in pest management and conservation Pages 181 198 in Stenseth N C and W Z Lidicker Jr Animal dispersal small mammals as a model Chapman and Hall New York New York USA Hastings A 1993 Complex interactions between dispersal and dynamics lessons from coupled logistic equations Ecology 7413621372 Krebs C J 1992 The role of dispersal in cyclic rodent populations Pages 160175 in Stenseth N C and W Z Lidicker Jr Animal dispersal small mammals as a model Chapman and Hall New York New York USA Larsen K W and S Boutin 1994 Movements survival and settlement of red squirrel FW662 Lecture 9 Immigration and Emigration 5 Tamiasciurus hudsom39cus offspring Ecology 75214223 Lidicker W Z 1975 The role of dispersal in the demography of small mammals Pages 103 128 in F B Golley K Petrusewicz and L Ryszkowski eds Small Mammals Their Productivity and Population Dynamics Cambridge University Press Cambridge United Kingdom Morris D W 1982 Agespeci c dispersal strategies in iteroparous species who leaves when Evolutionary Theory 65365 Porter J H and J L Dooley Jr 1993 Animal dispersal patterns a reassessment of simple mathematical models Ecology 7424362443 Ricklefs R E 1990 Ecology 3rd Edition Freeman New York New York USA 896 pp Sinclair A R E 1992 D0 large mammals disperse like small mammals Pages 229242 in Stenseth N C and W Z Lidicker Jr Animal dispersal small mammals as a model Chapman and Hall New York New York USA FW662 Lecture 10 Predation 1 Lecture 10 Predation Parasitism and Herbivory Reading Gotelli 2001 A Primer of Ecology Chapter 6 pages 125153 Renshaw 1991 Chapter 6 Predatorprey processes Pages 166204 Optional Gasaway W C R D Boertje D V Grangaard D G Kelleyhouse R O Stephenson and D G Larsen 1992 The role of predation in limiting moose at low densities in Alaska and Yukon and implications for conservation Wildlife Monograph 120 59 pp Boutin S 1992 Predation and moose population dynamics a critique Journal of Wildlife Management 56116127 Krebs C J et al 1992 What drives the snowshoe hare cycle in Canada39s Yukon Pages 886896 in D R McCullough and R H Barrett eds Wildlife 2001 Populations Elsevier Applied Science New York New York USA The notion that one trophic level controls the dynamics of another trophic level is central to ecology Energy ows between trophic levels The input of energy to a higher trophic level population from a lower trophic level population can be controlled by the amount of energy available in the lower level or by the amount of energy the higher trophic level is able to consume Hence the ideas of this chapter are as pertinent to herbivores and plants as they are to predators and their prey If the prey population limits the predator population the system is referred to as bottomup control Vice versa if the predator limits the prey population then the system is referred to as topdown control The idea that predators control prey Hairston et al 1960 or topdown control has been pervasive in the literature for a long time In a review of the impact of smallrodent population dynamics Hanski et al 2001 conclude that predators are indeed causing the regular multiannual population oscillations of boreal and arctic small rodents voles and lemmings Alternatively the notion that predators only take the weak and the sick Errington 19 suggests that predators are not controlling prey populations Rather predators are only taking the doomed surplus and therefore are not additive to the prey death rate but are compensatory Cause and Effect Demonstration of predators having an impact on prey population must be done via manipulative experiments The following graph shows a correlation between predator density and the proportion of prey consumed killed by predators However this correlation is not evidence that the predator population is in any way controlling the prey population density For example reduction in predator numbers might result in increased starvation of prey because of density dependence and hence predation is compensatory to starvation Unfortunately actual manipulations of predator and prey FW662 Lecture 10 Predation populations are limited and the few that do appear in the literature usually lack replication and appropriate spatial and temporal controls As a result this area of population dynamics is still highly controversial mungas Most of the predatorprey models stem either from the original models of Lotka 1925 and Volterra 1926 or from those of Nicholson 1933 and Nicholson and Bailey 1935 These models provides an historical perspective and a foundation on which to build a rigorous mathematical to base our discussion Both types of model are based on simple assumptions Both are a closed system involving coupled interactions Neither model involves age structure for either the predator or prey Predation is a linear function of prey density implying insatiable predators with not handling time And both models assume that the prey eaten are direcytly converted into new predators The chief difference lies in the LotkaVolterra models being differential equations while the NicholsonBailey models are based on difference equations giving discrete non overlapping generations Hassell and Anderson 1989 The following equations are the LotkaVolterra model 61N1 7 blN1 7 lel blNl 7 5N1N2 sz 7 szz T dzNz BN1N2 T dzNz The death rate 61 of the prey N 1 is set proportional to the size of the predator population N as 6N2 Likewise the birth rate b of the predator population is set proportional to the prey population as SN 1 Units of b1 b2 all and 612 are time39l and units for 5 are predators bompredatorpreytime and for 6 prey eatenpreypredatortime Thus the equations generate units of preytime or predatorstime for the respecitve rates Some of the important characteristics of these equations are lack of densitydependence death rate of prey only depends on predator and prey FW662 Lecture 10 Predation 3 death rate of predator only depends on predator and birth rate of predator depends on predator and prey LotkaVolterra equations predict a stable cycle of predator and prey for a pathological set of parameter values Hassell and Anderson 1989 whereas the difference equation formulation results in a expanding oscillations as shown in the following graph Deterministic Predator Prey 4 0 35 N1 Prey N2 Predator Populations 0 1 0 2 O 3 0 4 0 5 0 60 Time t Equilibrium can only be obtained via very carefully set parameter values Yet the literature is full of theoretical results derived from these simple equations More appalling these equations have been suggested as explaining some of the classic predatorprey cycles observed in nature e g the lynx cycle based on fur returns for the Mackenzie River region of Canada 1821 1934 reported by Elton and Nicholson 1942 These equations are a classic case of the mathematical tail wagging the biological dog The equations have so little reality that its hard to see how they have much to do with real systems Still numerous hypotheses have been developed from these equations I will now explore some of the extensions and the consequences of these modifications to the predicted behavior of the predator and prey populations Density Dependence As a first attempt to incorporate some biological realism density dependence of the prey population might be included As formulated above the prey population grows exponentially without predators Density dependence might also be incorporated into the predator population giving the following equations le 2 V blN1 7 SNIN2 7 clN1 sz 2 7 BNINZ 7 dzNz 7 czNz Cycles no longer result with these equations FW662 Lecture 10 Predation 4 Extensions commonly made to the LotkaVolterra model involve the functional and numerical responses The functional response de nes the changes in the per capita predation rate as prey density increases The numerical response de nes the changes in predator density as prey density increases Functional Response Curves A predator s functional response is its per capita feeding rate on prey Holling 1965 1966 suggested that the predator should not be able to consume an unlimited number of prey as the prey population increases That is in the LotkaVolterra equations the number of prey consumed per predator is unlimited as the prey population increases The number of prey removed is 5N 1N2 so that the number of prey eaten per predator is unlimited as N 1 increases to infinity Holling proposed 3 models of the rate of prey capture per predator as a function of prey population density Types 1 II and 111 Type I is the default case already modeled in the LotkaVolterra equations Type I m aTN 1 where m is number of prey eaten over a time period T per predator and a is a proportionality constant Type 11 m a TN 1 a THN where TH is the time required to catch and devour a prey handling time This equation is obtained by substituting into the Type I and correcting for the handling time m aT ijgN and solving for m Type 111 m a TNn 1 a TENT where n is generally set equal to 2 n generates a lag time in the learning curve of the predator or a quottraining effectsquot The idea of adding a power term is a common mathematical trick e g the Richard s equation described as a modification of the logistic population growth model For the following examples the parameter values are I a 05 T 1 II a 04 T 1 TH 03 and III a 0015 T 1 TH 03 and n 2 Functional Response Curves Prey Eatern per Predator per Time 0 5 1O 15 20 25 3O Prey Population FW662 Lecture 10 Predation 5 Depensatory mortality is a term used to describe the decrease in the rate of prey mortality as the prey population increases That is as all the predators become satiated because of the large numbers of prey the number of prey killed per number of prey available declines so that the prey survival rate will increase This phenomena will occur as the Type II and III curves reach asymptotic values of the prey population The LotkaVolterra model is extended to incorporate Type II and III curves ie the rate at which predators eat prey is modi ed by the functional responses shown above The cyclic behavior can persist with Type II and III functional responses incorporated into the LotkaVolterra equations Holling 1965 justified the Type III functional response curve based on empirical evidence He buried saw y cocoons in sand the prey and let mice Peromyscus search for them Until they learned how to find the cocoons they were less effective than after they had become proficient Swenson 1977 in Emlen 1984 108 constructed functional response curves for walleye and sauger and found that a Type II curve was adequate Skalski and Gilliam 2001 review the literature on functional response curves and presented statistical evidence from 19 predatorprey systems that three predator dependent functional responses BeddingtonDeAngelis CrowleyMartin and HassellVarley ie models that are functions of both prey and predator abundance because of predator interference can provide better descriptions of predator feeding over a range of predatorprey abundances No single functional response best described all of the data sets The BeddingtonDeAngelis functional response curve Beddington 1975 DeAngelis et al 1975 is N N aNl 1 2 1 bN1 cN2 71 where when the value of 0 becomes zero this functional response curve becomes identical to Hollings Type II curve This model assumes that handling and interfering are exclusive activities The CrowleyMartin Crowley and Martin 1989 allows for interference among predators regardless of whether a particular individual is currently handling prey or searching for prey The CrowleyMartin model thus adds an additional term in the denominator aN1 N 1 N2 1 le CN2 7 1 ch1N72 7 1 which can be simplified to aN1 7 1 bN11 cN2 7 139 As with the BeddingtonDeAngelis curve making 0 equal to zero results in a Type II curve The third functional response curve considered by Skalski and Gilliam 2001 was the HassellVarley Hassell and Varley 1969 model Nla N2 FW662 Lecture 10 Predation 6 CW1 le sz where when m becomes zero reduces to a Type II curve Np N2 Vucetich et al 2002 also consider a number of different functional response curves to model the relationship between wolves and moose on Isle Royale Predator density explained more variation in kill rate than did prey density R 2 036 vs R 2 017 respectively The ratiodependent model greatly outperformed the preydependent model Nevertheless the ratiodependent model failed to explain most of the variation in kill rate R 2 034 The ratio J r J 7 pr 3 J r J J should disappear as investigators recognize that both models are overly simplistic Numerical Response Curves Numerical response is the response of predator populations to prey populations Predator birth rate may be a function of the food intake rate so that increased prey availability may result in an increased birth rate up to some asymptotic value Another possibility is that predators immigrate to an area or aggregate in an area of high prey density Type II and III curves are also useful for modeling this process This type of response results in a numerical response of the predator to the prey ie the number of predators increases in response to the number of prey The following graph illustrates an hypothetical example YoungPredator Food Intake per Predator Maker 1970 in Emlen 1984 demonstrated a Type 111 response of jaeger nest density to lemming density Johnson and Carpenter 1994 used functional and numerical responses to develop a fishangler interaction model Stochasticity The cyclic behavior of these equations will not be retained in a model that includes demographic stochasticity Typically both populations will go extinct FW662 Lecture 10 Predation 7 What predatorprey system can you think of that does not have demographic quot F 39 39 39 quot quoty might also reasonably be added to the LotkaVolterra model ie the basic rate constants become functions of the environment For example wolves pursuing moose bene t from snow conditions that support the weight ofa wolf but not the weight of a moose An example of a LotkaVolterra model which incorporates demographic stochasticity is shown in the following graph 1 39 Stochastic PredatorPrey Model 0 O Prey 01 OO 0 Population Size M 8 A A O O 2 Time t Spatial stochasticity should also be added to provide a realistic model Examples of some of the hypotheses some authors consider the following as conclusions taken from Emlen 1984 l Predatorprey system is more likely stable if the predator is not highly efficient at finding and capturing prey 2 Predatorprey system is more likely stable if predator is not highly efficient in handling prey 3 Predatorprey system is more likely stable if predator is not highly efficient at converting food to growth and reproduction 4 Enrichment of a predatorprey system by addition of food for the prey or alternative sustenance for the predator destabilizes the system 5 Many most prey species have available to them some form of refuge that prevents extinction by the predator As a result the predatorprey system may show greater amplitude in the cycling 6 Time lags destabilize a predatorprey system 7 Densitydependence takes out cyclic behavior of a predatorprey system 8 Stochasticity both demographic and environmental takes out cyclic behavior of a predatorprey system FW662 Lecture 10 Predation 8 9 Spatial stochasticity can be viewed as a refuge system or as a metapopulation system Too low of prey dispersal makes the system unstable just as too high of prey dispersal does Coevolution The predatorprey system is sensitive to the efficiency of the predator taking prey and ultimately determines the stability of the system We can assume that natural selection is constantly operating on the system so that the predator is improving its abilities to capture prey whereas the prey is improving its abilities to avoid capture The predator and prey are coevolving like an evolutionary arms race Experimental Studies Huffaker experiments Huffaker 1958 Frey was sixspotted mite Eotetranychus sexmaculatus predator was a predatory mite Typhlodromus occidentalis Oranges rubber balls and wax paper were used to construct experiments where the prey had different habitat patches available to it providing different configurations of prey to the predator To ensure that his experiment was sufficiently complex Huffaker placed 40 oranges or balls in a 4 X 10 rectangular array on each of a number of adjacent trays Migration of predators across trays was restricted by inserting vaseline barriers whilst migration of prey over these barriers was achieved by providing each tray with wooden posts from which the prey could launch themselves on a silken thread aided by currents from an electric fan Renshaw 1991206 This is an example ofbiological modeling not using equations Huffaker did observe oscillations in the predator and prey system Wolves and Moose in Alaska and Yukon Gasaway et al 1983 1992 Boutin 1992 High predation on moose calves by wolves was thought to be limiting the moose population After wolves were removed from the system the moose populations responded upward Hence moose at low densities were being controlled by wolves Gasaway et al 1992 termed the original state of the system as a Low Density Dynamic Equilibrium LDDE When wolves were removed temporarily from the system the moose population climbed towards the carrying capacity set by browse and escaped the wolf predation limitation because relatively more moose calves escaped wolf predation Mathematics if any of LDDE Messier 1994 Is the reason that cycles are most often observed in Arctic systems because of the simplicity of these systems and the strong seasonality Wolves and Moose on Isle Royale McLaren and Peterson 1994 This article infers top down control because the only manipulation was a change in the wolf population Are their inferences really valid given the lack of a true experiment Coyotes and Mule Deer in Northwestern Colorado Bartmann et al 1992 These authors FW662 Lecture 10 Predation 9 observed high overwinter fawn mortality from coyote predation CB Fawn Survlval Coyote Study 1 Guyana Honlava 08 08 MQHMHHW VVIl39 BI39 Removal of coyotes 93 78 and 47 in 198586 8687 and 8788 respectively did not increase fawn survival compared to the previous 4 winters prior to coyote removal and suggested an increase in starvation of fawns Hence coyotes are providing compensatory mortality in the fawn population not additive mortality as would be suggested by the large numbers of fawns killed by coyotes each winter Predators only take weak and sick NOT but prudent predators do take the easiest prey Predatlon o StarvatIon Survlvad 44 o O O 40 i r 39 39 0quot 1 0 00 9 2 6 3 9 H o E 333 uquot g h39 32 139 U 0 a g m 5 3305 0 3 I quot O 29 33 5 O O 34 o 20 0 0 an 40 so so 1oo1ao14o1oo1eo oo DayIISquad FW662 Lecture 10 Predation 10 Mule deer at KCC Snowshoe hare cycles Krebs et al 1992 Krebs et al 1992 present good evidence of snowshoe hare Lepus americanus cycles in southwestern Yukon Four hypotheses have been proposed to explain the hare cycles 1 Keith hypothesis winter food shortage reduces reproduction at the population peak and causes starvation losses which start the cyclic downturn and predation which continues the downturn and reduces hares to low numbers 2 Plant chemistry hypothesis qualitative nutritional changes in the hare s food plants 3 Predation hypothesis differs from Keith s hypothesis by being a singlefactor model for the hare cycle 4 Chitty hypothesis or polymorphic behavior hypothesis driving mechanism is the spacing behavior of the hares themselves In their study at Kluane Lake during 197784 they have performed experiments with winter food supplementation of hares The overall results of these studies were rather inconsistent with the food hypothesis Winter food shortage was not necessary for the cyclic decline and extra food would neither slow the rate of decline of the hares nor delay in all populations the start of the decline In these experiments we did not control for increased predation on the foodsupplemented areas Krebs et a1 1992888 Keith et a1 1984 studying the same cyclic decline in central Alberta hares also concluded that food shortage was not present on all areas where hares were declining If these results are accepted neither the original Keith hypothesis nor the plant chemistry hypothesis can be supported as an explanation of cyclic events on a local scale Krebs et a1 1992888 Krebs et a1 1992 eliminate the Chitty hypothesis because no evidence of social mortality either directly through infanticide or ghting or indirectly through territoriality and dispersal Through the process of elimination the only hypothesis left is predation From 1986present they have set up experiments that provide controls fertilization plots food supplementation predatorproof fences which do not eliminate avian predators and foodpredator fence Results as of 1992 suggested food supplementation had increased April 1 populations but they had not yet observed the downswing in the hare cycle Presumably eliminating predators would keep the hare population high if predation was the mechanism that causes the down turn Largemouth bass in a Michigan lake Mittelbach et al 1995 This paper presents the results of a longterm study of changing predator densities and cascading effects in a Michigan lake in which the top carnivore the largemouth bass Micropterus salmoides was eliminated in 1978 and then reintroduced in 1986 The elimination of the bass was followed by a dramatic increase in the density of planktivorous fish the disappearance of large zooplankton e g two species of Daphnia that had historically dominated the zooplankton community and the appearance of a suite of smallbodied FW662 Lecture 10 Predation 11 cladoceran zooplankton species The system remained in this state until bass were reintroduced As the bass population increased the system showed a steady and predictable return to its previous state planktivore numbers declined by two orders of magnitude largebodied Daphnia reappeared and again dominated the zooplankton and the suite of smallbodied cladocerans disappeared Within each cladoceran species there was a steady increase in mean adult body size as planktivore numbers declined Total zooplankton biomass increased 710fold following the return of largebodied Daphnia and water clarity increased signi cantly with increases in Daphnia biomass and total cladoceran biomass These changes in community structure and trophiclevel biomasses demonstrate the strong impact of removing a single keystone species and the capacity of the community to return to its previous state after the species is reintroduced Competition for shelter space causes densitydependent predation in damselfishes Holbrook and Schmitt 2002 demonstrated that competition for shelter spaces caused more predation in damselfrshes These species shelter in branching corals or anemones and when refuge spaces to protect them were lled with their intra specifrc competitors more mortality from predation was found Cycles Post et al 2002 suggest that wolves cause moose population on Isle Royale island in Lake Superior to cycle Although the data do suggest that moose populations have peaked twice in the interval 19581998 claiming that wolf r r 39 quot are r quot 39 is not A f quot 39 Rather there does appear to be a correlation between these populations However many different variables particularly weather variables could be found to correlate with these populations and most investigators would not claim a causeandeffect relationship had been found Literature Cited Bartmann R M G C White and L H Carpenter 1992 Compensatory mortality in a Colorado mule deer population Wildlife Monograph 121 139 Beddington J R 1975 Mutual interference between parasites or predators and its effect on searching efficiency Journal of Animal Ecology 51331340 Boutin S 1992 Predation and moose population dynamics a critique Journal of Wildlife Management 56116127 Crowley P H and E K Martin 1989 Functional responses and interference within and between year classes of a dragon y population Journal of the North American Benthological Society 8211221 DeAngelis D L R A Goldstein and R V O Neill 1975 A model for trophic interaction Ecology 56881892 FW662 Lecture 10 Predation 12 Elton C and M Nicholson 1942 The tenyear cycle in numbers of the lynx in Canada Journal of Animal Ecology 11215244 Errington P 19 Gasaway W C R O Stephenson J L Davis P E K Shepherd and O E Burris 1983 Interrelationships of wolves prey and man in interior Alaska Wildlife Monograph 84 50 pp Gasaway W C R D Boertje D V Grangaard D G Kelleyhouse R O Stephenson and D G Larsen 1992 The role of predation in limiting moose at low densities in Alaska and Yukon and implications for conservation Wildlife Monograph 120 59 pp Hairston N G F E Smith L B Slobodkin 1960 American Naturalist 94421 Hanski I H Henttonen E Korpimaki L Oksanen and P Turchin 2001 Smallrodent dynamics and predation Ecology 8215051520 Hassell M P and R M Anderson 1989 Predatorprey and hostpathogen interactions Pages 147196 in J M Cherrett ed Ecological concepts Blackwell scienti c Publ Oxford United Kingdom Hassell M P and G C Varley 1969 New inductive population model for insect parasites and its bearing on biological control Nature 22311331136 Holbrook S J and R J Schmitt 2002 Competition for shelter space causes density dependent predation mortality in damselflshes Ecology 8328552868 Holling C S 1965 The functional response of predators to prey density and its role in mimicry and population regulation Memorandum Entomological Society of Canada No 45 Holling C S 1966 The functional response of invertebrate prey to prey density Memorandum Entomological Society of Canada 48 148 Huffaker C B 1958 Experimental studies on predation dispersion factors and predatorprey interactions Hilgardia 27343383 Johnson B M and S R Carpenter 1994 Functional and numerical responses a framework for fishangler interactions Ecological Applications 4808821 Keith L B J R Cary O J Rongstad and M C Brittingham 1984 Demography and ecology ofa declining 39 hare r 39 quot Wildlife u l 90 1 FW662 Lecture 10 Predation 13 Krebs C J et al 1992 What drives the snowshoe hare cycle in Canada39s Yukon Pages 886 896 in D R McCullough and R H Barrett eds Wildlife 2001 Populations Elsevier Applied Science New York New York USA Lotka A J 1925 Elements of Physical Biology Williams and Wilkins Baltimore Reissued as Elements of Mathematical Biology by Dover 1956 Maker W J 1970 The pomarine jaeger as a brown lemming predator in northern Alaska Wilson Bull 82130157 McLaren B E and R 0 Peterson 1994 Wolves moose and tree rings on Isle Royale Science 26615551558 Messier F 1994 Ungulate population models with predation a case study with the North American moose Ecology 75478488 Mittelbach G G A M Turner D J Hall J E Rettig and C W Osenberg 1995 Perturbation and resilience a longterm wholelake study of predator extinction and reintroduction Ecology 762347 2360 Nicholson A J 1933 The balance of animal populations Journal of Animal Ecology 2 131 Nicholson A J and V A Bailey 1935 The balance of animal populations Part 1 Proceedings ofthe Zoological Society of London 1935551598 Post E N C Stenseth R 0 Peterson J A Vucetich and A M Ellis 2002 Phase dependence and population cycles in a largemammal predatorprey system Ecology 832997 3002 Skalski G T and J F Gilliam 2001 Functional responses with predator interference viable alternatives to the Holling Type 11 model Ecology 8230833092 Swenson W A 1977 Food consumption of walleye Stizostedion vitreum vitreum and sauger S canadense in relation to food availability and physical conditions in Lake of the Woods Minnesota Shagawa Lake and western Lake Superior Journal Fisheries Research Board of Canada 34 16431654 Volterra V 1926 Variazioni e uttuazioni del numero d individui in specie animali conviventi Memorie della R Accademia Nazionale dei Lincei 231113 Translation in Chapman R N 1931 Animal ecology pages 409448 McGrawHill New York New York USA FW662 Lecture 10 Predation Vucetich J A R 0 Peterson and C L Schaefer 2002 The effect of prey and predator densities on wolfpredation Ecology 8330033013 14 FW662 Lecture 7 Compensatory mortality 1 Lecture 7 Additive vs compensatory mortality and MSY Reading Nichols J D M J Conroy D R Anderson and K P Burnham 1984 Compensatory Mortality in waterfowl populations a review of the evidence and implications for research and management Transactions of North American Wildlife and Natural Resources Conference 49535554 Optional Nichols J D 1991 Responses of North American duck populations to exploitation Pages 498525 in C M Perrins JD Lebreton and G J M Hirons eds Bird Population Studies Oxford New York New York USA Smith G and R Reynolds 1992 Hunting and mallard survival Journal of Wildlife Management 56306316 Sedinger J S and E A Rexstad 1994 Do restrictive harvest regulations result in higher survival rates in mallards A comment Journal of Wildlife Management 58571577 Smith G and R Reynolds 1994 Hunting and mallard survival a reply Journal of Wildlife Management 58578581 Clark W R 1987 Effect of harvest on annual survival of muskrats Journal of Wildlife Management 51265272 I will illustrate the concept of compensatory mortality with a simple example Assume that 90 animals start the biological year All harvest takes place before any natural mortality occurs following the assumptions of Boyce et a1 1999 Further assume that the natural mortality occurs in densitydependent fashion ie survival from the end of the harvest period to the start of the next year is defined as Snl307131N where Sn is the survival from the end of the harvest period to the start of the next year and let 50 08333 and 51 00055556 This function is plotted on the following graph along with the densityindependent situation where no response in survival is allowed as a function of population size These lines are labels compensatory for density dependence and additive for density independence because these are the underlying assumptions that result in compensatory and additive mortality FW662 Lecture 7 Compensatory mortality 2 Survival vs Population Size 09 08 D 507 D g 06 305 04 03 0 20 4O 60 80 100 Population Size Assume now that for the base situation 13 of the 90 animals be removed by hunting so that for the 60 left Sn 08333 0005555660 05 under the assumption of density dependence Thus 30 of these animals survive the year Survive Natural Fate Harvest Now we want to manipulate the system by removing the hunting mortality ie let the harvest rate equal zero Under the assumption of a densitydependent response to the FW662 Lecture 7 Compensatory mortality 3 removal of hunting 90 animals undergo natural mortality and the survival rate is Sn 08333 0005555690 03333 Thus only 30 animals survive the year just as in the case of hunting mortality of 33 Survive Natural Fate Harvest The hunting mortality is compensated for by an increase in survival of the animals remaining after the hunting season by the densitydependent decrease in mortality because of fewer animals present in the population The overall survival rate for the year S with no subscript is de ned as a function of the harvest rate h and the survival rate after the hunting season The overall survival rate will be the product of the survival through the hunting season 1 h and S For the case where mortality is density dependent ie Sn is a function of density S 1 hBo 51W hN If we graph the overall survival rate S we get the relationship FW662 Lecture 7 Compensatory mortality 4 Harvest Rate vs Annual Survival Rate 1 08 O 0 Survival Rate 0 in O N 0 02 04 06 08 1 Harvest Rate This curve of compensation is relatively at for quite a range of harvest rates because the natural survival rate compensates for the increase in harvest rate by increasing because of the decreasing number of animals in the population The maximum overall survival is obtained at 2ND1 7 50 2N5 h 1 For the values of 50 08333 51 0005556 andN 90 the maximum survival is obtained at a harvest rate of h 016667 If the hunting mortality had been additive then the survival rate after hunting observed for the 60 animals in the base situation would continue to apply to 90 animals so that 45 would survive the year This situation is demonstrated in the following histogram and is illustrated in the above plot by the line labeled additive No response in the natural mortality rate is available to compensate for increased harvest so the additive line decreases linearly in response to an increase in the harvest rate FW662 Lecture 7 Compensatory mortality 5 Survive Natural Fate Harvest Another common misconception about our example is that if the harvest is removed all the harvested animals will live giving the following result This result I label super additive To achieve this response in a population you would have to have reverse densitydependence ie the natural mortality rate would have to decrease as the population increased Survive Natural Fate Harvest Anderson and Burnham 1976 presented a mathematical argument for compensatory mortality They derived their results based on instantaneous rates of harvest and natural mortality The example above is based on nite rates with the assumption of no natural mortality during the harvest period For nite rates and no natural FW662 Lecture 7 Compensatory mortality 6 mortality during the hunting season their additive mortality results are the same straight line graph as shown above However if some natural mortality occurs during the hunting season the line deviates below the straight line shown above Additive Mortality AmalSnrvivalRareS Hunting Mmmnynm K Under the compensatory mortality hypothesis with density dependence operating on survival rate after the hunting season Anderson and Burnham 1976 present the following graph The shape and general conclusions reached from this graph are the same as illustrated above Over some range of harvest 0 to c the annual survival rate remains unchanged in response to harvest However beyond the threshold value of harvest c the densitydependent response of the population cannot compensate for the harvest so the annual survival rate declines Compensatory Mortality AnnualSurvivalRmS Hun ngMorulitanteK The natural mortality function to generate such a survival function in response to hunting mortality is the following The population identified with 0 corresponds to the population size at the threshold in the above graph The xaXis is the posthunt population size and the yaXis is the mortality rate from postharvest to the start of the next year Any FW662 Lecture 7 Compensatory mortality 7 population harvested at a rate greater then c has no natural mortality following harvest thus illustrating complete compensation Natural Mortality vs Population Size 05 Mortality Rate 0 O O M 0 A O 0 Population Size Three approaches have been used to test between the 2 hypotheses Regression of Squot vs Ki where K is kill rate not carrying capacity Sampling covariance of the 2 estimates SAquot and 15 induces a negative relationship Burnham and Anderson 1979 This covariance must be removed to compute a proper test of these 2 quantities Splitting raw data in half Nichols and Hines 1983 is one approach to removing the covariance Half the data are used to estimate Squot and the other half to estimate Ki Both hypotheses in a single equation Burnham et al 1984 SI S01 7 bKl H0 b 0 means compensation Ha 0 lt b lt 1 means partial compensation Ha b 1 means additive Continuity of compensatory and additive hypotheses Relation of survival to population size or harvest Instantaneous vs nite representations N NOeXpb 7 m0 no 7 m0n0t where m0 is shing mortality in the absence of natural mortality and n0 is natural mortality in the absence of shing mortality This equation assumes additive mortality The term mon0 just speci es that a sh cannot die from both natural and shing mortality In reality m0 can never be measured see Anderson and Burnham 1981 The parameters m and n are actually measured so that overall mortality is m n which conceptually is not equal to m0 no 7 mono For compensatory mortality 71 must be made a function of m FW662 Lecture 7 Compensatory mortality 8 Another example for the nite time model of how compensation can be significant assumes that densitydependent mortality m ie the mortality rate for the period post harvest until the start of the next year is modeled by the following function m 7 expBoBN 7 11W or equivalently survival as a function of density S 7 1 7 expBoBN 7 11W Plug the values 50 179175 51 22E8 and 6 4 into this function The resulting curve as a function of N with h 0 looks like this 08 07 06 05 04 03 Mortality Rate m 02 01 0 20 40 60 80 100 Population Size N Plugging this densitydependent mortality curve into the expression for overall survival ie the product of survival through the harvest period and the survival through the period from the end of harvest until the start of the next year Sn gives S 1 h1eXPi50i51N hMel and results in a curve of survival rate as a function of harvest rate like the following FW662 Lecture 7 Compensatory mortality 9 06 05 04 03 02 Survival Rate 8 01 O 02 04 06 08 1 Harvest Rate h In other words as one of my game warden friends says you gotta shoot m to save m A harvest rate of about 02 results in the maximum number of animals at the end of the winter far more than if harvest is zero With this mortality function you can harvest up to just more than 60 of the population and still have the same number of animals left at the end of the year as you would have with no harvest Examples Waterfowl Bumham and Anderson 1984 Bumham et a1 1984 Nichols et a1 1984 Smith and Reynolds 1992 Sedinger and Rexstad 1994 Smith and Reynolds 1994 Muskrats Clark 1987 Mule deer Bartmann 1992 Discussion Why have so many studies examined reproduction in response to population size but not survival rates Literature Cited Anderson D R And K P Bumham 1976 Population ecology ofthe mallard VI The effect of exploitation on survival Resoure Publication 128 U S Fish and Wildlife Service 66 PP FW662 Lecture 7 Compensatory mortality 10 Anderson D R and K P Burnham 1981 Bobwhite population responses to exploitation two problems Journal of Wildlife Management 4510521054 Bartmann R M G C White and L H Carpenter 1992 Compensatory mortality in a Colorado mule deer population Wildlife Monograph 121 139 Boyce M S A R E Sinclair and G C White 1999 Seasonal compensation of predation and harvesting Oikos 87419426 Burnham K P and D R Anderson 1984 Tests of compensatory vs additive hypotheses of mortality in Mallards Ecology 65 1051 12 Burnham K P G C White and D R Anderson 1984 Estimating the effect of hunting on annual survival rates of adult mallards Journal of Wildlife Management 48350361 Clark W R 1987 Effect of harvest on annual survival of muskrats Journal of Wildlife Management 51265272 Nichols J D and J E Hines 1983 The relationship between harvest and survival rates of mallards a straightforward approach with partitioned data sets Journal of Wildlife Management 47334348 Nichols J D M J Conroy D R Anderson and K P Burnham 1984 Compensatory Mortality in waterfowl populations a review of the evidence and implications for research and management Transactions of North American Wildlife and Natural Resources Conference 49535554 Sedinger J S and E A Rexstad 1994 D0 restrictive harvest regulations result in higher survival rates in mallards A comment Journal of Wildlife Management 58571577 Smith G and R Reynolds 1992 Hunting and mallard survival Journal of Wildlife Management 56306316 Smith G and R Reynolds 1994 Hunting and mallard survival a reply Journal of Wildlife Management 58578581 FW662 Lecture 12 Genetics 1 Lecture 12 Natural Selection and Population Regulation Reading Shields W M Chapter 8 Optimal inbreeding and the evolution of philopatry Pages 132159 in I R Swingland and P J Greenwood eds The ecology of animal movement Clarendon Press Oxford United Kingdom Meffe G K 1986 Conservation genetics and the management of endangered shes Fisheries 111 1423 Optional Hedrick P W and P S Miller 1992 Conservation genetics techniques and fundamentals Ecological Applications 23046 Lande R and G Barrowclough 1987 Effective population size genetic variation and their use in population management Pages 87123 in M E Soule ed Viable populations for conservation Cambridge University Press New York New York USA Jimenez J A K A Hughes G Alaks L Graham and R C Lacy 1994 An experimental study of inbreeding depression in a natural habitat Science 266271273 Genetic models are much like the predatorprey and competition models we have been studying they appear to be very naive and we don39t have the hard data to support their predictions Most of the following material on methods and models comes from Meffee 1986 Measuring genotypes in populations Different classes of genes blood group antigens proteins restriction sites on mitochondrial DNA mtDNA nuclear DNA Electrophoresis measures differences in proteins that re ects genotype Loci particular protein being measured Allele specific types of protein found at the loci usually denoted as A and a Each loci in an individual has 2 alleles one from father and one from mother These 2 alleles proteins can be combined in an individual as Ad ad However a loci across the population may have more thanjust 2 alleles eg A B and C Then individuals in the population may have any of the following combinations AA AB AC BB BC and CC Heterozygosity is defined as the variation in a loci ie frequency of Act allele pair for alleles A and a Suppose Probability ofA is PrA p 03 Then FW662 Lecture 12 Genetics 2 PrAA PrA2 032 009 Homozygous Praa lPra2 072 049 Homozygous PrAa PrAlPra X 2 Heterozygous 07 X 03 X 2 042 Total 10 HardyWeinberg law implies these probabilities ie random mating in a panmictic mixed population Heterozygosity of loci j in a population is measured as y 2njl 7 2 x5 1 2nj 7 1 where n is number of animals measured at loci j so 271 is the number of alleles measured and x1 is frequency of allele Iat locij The term 2nJ2nj l is a small sample bias correction For PrA 03 x11 03 and x2 07 so h 042 which is the probability ofthe loci containing 2 unlike alleles at loci j Homozygosity is the probability of 2 like alleles at loci j and thus equals 1 h Random mating means each animal has equal probability of breeding with any other animal in the local population known as a deme A deme is the unit within a population where mating is random Two segments eg north and south of a population might each comprise a deme At time 0 PrA p 03 The number ofA s in the population is 2np ie 2 alleles in each member of the population What happens after one generation in the population to the gene frequency of A Possible matings their probability of occurring and the probabilities of the offspring genetic frequencies are shown in the following table for PrA p 03 and Pra l p 07 FW662 Lecture 12 Genetics Parents Probability of Offspring s Offspring Combined Alleles Parents Mating Alleles Probability Probability act X cm 0492 aa 1 0492 aaXaA 049X042X2 cm 05 049X042 aA 05 049 X 042 aaXAA 049X009X2 aA l 049X009X2 aA X aA 0422 cm 025 025 X 0422 aA 05 05 X 0422 AA 025 025 X 0422 aAXAA 042X009X2 aA 05 042X009 AA 05 042 X 009 AA XAA 0092 AA 1 0092 Totals 100 100 At time 1 EA 2np and VarA 2np1 p The actual variation in p is Varp1 VarA2n l2n2 VarA p1 p2n At time 2 Varp2 p1 p1 1 12n2 In general at time t Varp VarpHl 7 l p1p 2n Increasing 71 decreases the rate of increase in variance and increasing p up to 05 increases rate of increase of variance FW662 Lecture 12 Genetics 4 Variance of allele frequency 025 2 02 E10 p03 3015 n20 p03 E 01 n50 p03 gtu005 110 F n10 p02 O Time Conservation implications of this graph are important The rst implication is that the smaller the population size the more chance of losing an allele and hence genetic variation or heterozygosity Second is that the probability of losing an allele increases as allele frequency goes down This random variation is call random drift In populations we are concerned about the overall level of heterozygosity in the population across all loci ForL loci Effective population size NE assumes random mating For most of the populations we are interested in we do not see random mating because the population consists of males and females The effective size of the population is a function of the number of males and females 1 Sex ratios NE 4 NM N Nm N FW662 Lecture 12 Genetics 5 0N Sex Ratio amp Ne cquot3100 80 m D 60 gt 39H 40 41 8 20 x 94 Ed 0 Males As sex ratio is biased away from 05 the effective population size declines drastically This result is because only a small portion of the population is responsible for 12 of the alleles in the next generation This has implications for male only hunting seasons 2 Distribution of progeny NE 4 N 2 O2 where 0218 the variance of the progeny distribution in the population Suppose that 1000 males mate with 1000 females and each female has on average 2 offspring that contribute to the next generation If the distribution of offspring is Poisson distributed then the mean equals the variance so 02 2 and NE 4 X 20002 2 2000 In contrast if 999 females each have 1 offspring and 1 has 1001 then 02 316 and Ne 4 X 20002 316 238 An example of this phenomena is the reintroduction of peregrine falcons on the east coast of the US Temple Pers Commun 95 of the present birds are descendants of 5 individuals of which 2 pairs are siblings all from Alaska Even though 16 subspecies were used for the introduction including individuals from Europe the Alaska birds provided the starting F1 generation FW662 Lecture 12 Genetics Progeny Distribution and Ne 2000 810 1500 1000 quot Inca11 Pop U39I O 0 5 10 15 20 25 30 35 Variance of Progeny Nunney and Elam 1992 suggest underestimates of effective population size are because of a failure to account for a long maturation time and problems with the correction for overlapping generations 3 Population Fluctuations When populations uctuate through time all the genetic variation for all future populations is contained in only a few survivors assuming the mutation rate is zero The harmonic mean population size represents the effect population size for the population 1 l l t N1 N2 N i Ne FW662 Lecture 12 Genetics 7 Pop Fluctuation amp Ne 0 I 1 N 100 In e Q Genera ti on The phenomena of a population dropping to a low level and then recovering is called a population bottleneck When populations go through a bottleneck the amount of genetic variation left after the bottleneck depends on 1 the size of the population in the bottleneck and 2 the length of time number of generations the population was in the bottleneck The percent of genetic variance remaining in the population after t generations is I 1 i 2N Bottleneck Effects Variance Remaining O h 10 100 Bottleneck Population Size FW662 Lecture 12 Genetics 8 Variance remaining 9 9 9 4 O 00 Variance Remaining 9 m Q Q 10 20 30 40 50 Time These graphs would suggest that severe bottlenecks are a complete disaster for genetic variation in the population We have to examine the 2 major assumptions we are making to arrive at this conclusion Bottlenecks do lower genetic variation or could the model be incorrect Bryant et al 1986 measured the effects of passing house ies through population bottlenecks Lewin 1987 Genetic effects that they saw in populations of ys that bred from 1 4 and 16 malefemale pairs in 3 separate experiments was an increase in variance not a decrease as most mathematical models of bottlenecks would imply There was more variation in the ys39 physical characteristics wing size and shape 8 traits in postbottleneck r r 39 quot than pr 39 r r 39 They measured 8 traits on 3000 ies From the formula for the percent of genetic variance retained we calculate that 1 pr expect 75 of original variation 2 pr expect 94 of original variation l6pr expect 98 of original variation Most variation came through the bottlenecks of intermediate sizes ie 4 and 16 pairs quotThe dogma of bottleneck theory has always assumed that the newly founded population is somehow at risk because of predicted lower variancequot Bryant Another explanation of why the observed reduction in variance is not as great as predicted by the above equation concerns frequency of alleles in the original population Suppose 200 alleles exist at a loci in the original population If only a single pair comes through a bottleneck at most 4 ofthese 200 alleles will exist However in the original population the 200 alleles are not equally frequent and hence do not contribute equally to the next generation Thus the reduction from 200 to 4 is a major reduction but not as much as it first appears I I FW662 Lecture 12 Genetics 9 Example of this dogma are cheetahs O39Brien et al 1985 1987 Lewin 1987 Caro and Laurenson 1994 Cheetahs prior to 10000 years ago were widely distributed Currently the entire population is more genetically uniform than inbred laboratory mice These animals are so genetically similar that skin graphs from one individual can be given to a completely unrelated individual Breeding success in captivity is poor with very low quality of the male39s spermatozoa and high infant mortality The usual explanation of these observations is a long and persistent bottleneck Similar characteristics are observed in the Florida panther Caro and Laurenson 1994 suggest genetics are not the problem They found that most cub mortality was from predation mainly hyeanas Reproduction in the wild was high compared to studies in captivity Reproductive data from captivity generally is highly variable with some zoos having good success This result suggests that breeding in captivity is affected by animal husbandry with inappropriate social conditions causing the low reproductive success and not genetics However Jimenez et al 1994 experimentally demonstrate that inbreeding reduces survival and hence fitness But is the treatment Jimenez et al applied a realistic manipulation ie severe inbreeding is known to be deleterious but does that imply that lower heterozygosity is necessarily as deleterious The second assumption we must question is the evidence for heterozygosity increasing fitness Genetic variation heterozygosity varies greatly from taxon to taxon and its evolutionary meaning is controversial What are the costs and bene ts to maintaining heterozygosity The loss of some alleles may be very beneficial ie if the allele is detrimental to survival under certain conditions Localized selection means that alleles beneficial to local situations are desirable and needed Other alleles may be neutral or negative so their loss may be neutral or even positive High heterozygosity may be detrimental to local conditions Shields 1983 Being a jack of all trades means you are a master of none Too much heterozygosity results in outbreeding depression The opposite phenomena is hybrid vigor Rhymer and Simberloff 1996 discuss the implications of increasing heterozygosity through hybridization Either outbreeding depression or hybrid vigor may result However the pure genetics of both populations may be lost Even if the genetics of the populations are not changed the population dynamics may be changed because the production of no offspring or infertile offspring lower the recruitment rate of one or both populations This article provides a great deal of practical advice on hybridization and introgressz39on gene ow between populations whose individuals hybridize achieved when hybrids FW662 Lecture 12 Genetics 10 backcross to one or both parental populations Inbreeding is often cited as evidence of the importance of heterozygosity ie inbreeding depression is bad so outbreeding must be good However inbreeding really supports the case that too much heterozygosity can be bad Inbreeding effects are often the result of 2 deleterious recessive alleles coming together Inbreeding can be used to purge a population of these deleterious alleles Breeding programs for domestic livestock and dogs are examples You save the good ones and throw the rest away The good ones are really good and the bad ones are really bad Evidence for localized selection or at least localized differences in genotypes Changes in allele frequency can occur over very short distances in populations We observed differences in allele frequencies between trap sites at Little Hills in distances lt5 km Populations are genetically structured with genetically diverse units Rhodes and Smith 1992 Hence genetic variation may appear to be high across the population but may be lower within the individuals that make up the population This argument suggests the random mating assumptions is not realistic Evidence that heterozygosity improves tness is strictly correlative See Table 1 from Rhodes and Smith 1992 Multiple correlations are run against H and some are bound to be found signi cant Second these correlations are looking at the results of selection not the process itself Frequencies of heterozygous loci suggest that too much heterozygosity is bad The following data are from Piceance mule deer Piceance Mule Deer 26 Loci 200 Frequency l l o 01 a Q 01 O 001234560123456 Heterozygous Loci The frequencies represent what is left and the possibility that these frequencies are correlated with fitness cannot be dismissed If linear correlations are conducted against fitness with increasing fitness up to the mode plus 1 and sharply decreasing fitness for gtH a positive correlation FW662 Lecture 12 Genetics 11 of H and tness would result However this correlation is spurious because we have not included the 0 tness animals no longer present in the population The problem with taking the sample from the population is that the low heterozygosity animals are overrepresented and hence a linear trend appears to be supported If equal sample sizes of each level of heterozygosity were used a quadratic relationship would likely be observed because there is some optimum level of heterozygosity Implications of genetics for metapopulations Patch extinctions mean you loose genetic diversity from the patch Recolonizers will effectively be coming through a bottleneck so that patch genetic diversity is now low As this process continues through time the genetic diversity of the entire population becomes very low Gilpin 1991 McCauley 1991 Hence if the metapopulation model is valid this loss of genetic variation cannot take place Either the metapopulation model is incorrect or bottleneck theory is wrong or heterozygosity is not all that important An example of such a population would be annual weeds Of courss you could argue that annual weeds are really one big r r 39 quot and not a r r 39 im The r r 39 quot in my garden hasn t gone extinct recently Implications of this idea are important to the SLOSS debate A single large reserve would have a larger effective population size whereas several small reserves has a lower effective population size because of the lack of a large panmictic population with random breeding Importance of intraspecific variation is discussed by Behnke 1992 For many trout populations the genetic analyses suggest they are the same genetically Yet they breed at different times of the year and have other important behavioral characteristics that allow their survival in unique environments Retention of the intraspecific genetic variation is necessary to maintain biodiversity Note that not all phenotypic characteristics are genetically based Kroodsma and Canaby 1985 found that marsh wren Cistothorus palustris song repertoire and style of delivery was genetically controlled In contrast James 1983 showed that a significant proportion of redwinged black bird Agelaius phoeniceus nestling development characteristics were not genetically based In both of these studies birds were moved as eggs to different environments to test the hypothesis of geneticbased versus environmentbased origin of the traits studied Conclusions about the importance of genetics in populations Genetic models need a great deal more testing Experiments are needed to test if heterozygosity is strongly related to fitness ie causeandeffect experiments M Smith Pers Comm conducted experiments with mosquito fish in wading pools Founding pairs were selected to give low or high heterozygosity with population persistence expected to be the greatest for high heterozygosity pools Have not heard the results Literature Cited Behnke R J 1992 Native trout of western North America American Fisheries Society FW662 Lecture 12 Genetics 12 Monograph 6 l Bryant E S McCommas and L Combs 1986 The effect of an experimental bottleneck upon quantitative genetic variation in the house y Genetics 114 1 191121 1 Caro T M and M K Laurenson 1994 Ecological and genetic factors in conservation a cautionary tale Science 263485486 Hedrick P W and P S Miller 1992 Conservation genetics techniques and fundamentals Ecological Applications 23046 James F C 1983 F 39 39 ofiuml39 39 39 39 diff quot quot in birds Science 221184186 Jimenez J A K A Hughes G Alaks L Graham and R C Lacy 1994 An experimental study of inbreeding depression in a natural habitat Science 266271273 Kroodsma D E And R A Canady Differences in repertoire size singing behavior and associated neuroanatomy among marsh wren populations have a genetic basis Auk 102439446 Lande R and G Barrowclough 1987 Effective population size genetic variation and their use in population management Pages 87124 in M E Soule ed Viable populations for conservation Cambridge University Press New York New York USA Lewin R 1987 The surprising genetics of bottlenecked ies Science 23513251327 McCauley D E 1991 Genetic consequences of local population extinction and recolonization TREE 658 Meffe G K 1986 Conservation genetics and the management of endangered fishes Fisheries 1111423 Nunney L and D R Elam 1992 Estimating the effective population size of conserved populations Conservation Biology 8 175184 O Brien S J M E Roelke L Marker A Newman C A Winkler D Meltzer L Colly J F Evermann M Bush and D E Wildt 1985 Genetic basis for species vulnerability in the cheetah Science 227 14281434 O Brien et al 1987 East African cheetahs evidence for two population bottlenecks Proceedings ofthe National Academy of Science USA 84508 FW662 Lecture 12 Genetics 13 Rhodes 0 E Jr and M H Smith 1992 Genetic perspectives in wildlife management the case oflarge herbivores Pages 985996 in D R McCullough and R H Barrett eds Wildlife 2001 Populations Elsevier Applied Science New York New York USA Rhymer J M and D Simberloff 1996 Extinction by hybridization and introgression Annual Review of Ecology and Systematics 2783109 Shields W M 1983 Optimal inbreeding and the evolution ofphilopatry Pages 132159 in I R Swingland and P J Greenwood eds The ecology of animal movement Clarendon Press Oxford United Kingdom FW662 Lecture 6 Evidence of density dependence Lecture 6 Mechanisms and evidence for density dependence Reading Sinclair A R E 1989 Population regulation in animals Pages l9724l in J M Cherrett ed Ecological concepts Blackwell Scienti c Publishers Oxford United Kingdom Optional Hixon M A S W Pacala and S A Sandin 2002 Population regulation historical context and contemporary challenges of open vs closed systems Ecology 8314901508 By de nition a population is regulated if it persists for many generations with ucuations bounded above zero with high probability Regulation thus requires densitydependent negative feedback whereby the population has a propensity to increase when small and decrease when large Hixon et al 2002 As stated by Hixon et al 2002 population regulation requires density dependence Density dependence need not be omnipresent to regulate a population Wiens 1977 but is essential at some time and place for longterm persistence Hassell 1986 What are the mechanisms that generate density dependence The possible causes of density dependence can be grouped into 2 categories competition and predation Competition for actually or potentially limiting resources bottom up regulation is always density dependent by de nition be it Via interference a direct interaction or exploitation an indirect interaction Hixon et al 2002 Predation broadly including disease parasitism parasitoids and herbivory is not always density dependent For predators to cause topdown regulation Via densitydependent prey mortality they must have a regulating total response which is the combination of a numerical response in predator population size a functional response in the per capita consumption rate and other behavioral and developmental responses to changes in prey abundance Hixon et al 2002 and references therein Mechanisms of density dependence Must have the rates of births deaths immigration or emigration change relative to population size FW662 Lecture 6 Evidence of density dependence 2 Density dependent Rates Birth Rate H U1 Births per caplta O in H Deaths per caplta Death quot02 Rate 0 IlIIIlIIIIllIIlIlIIIlO O 20 40 6O 80 100 Nt Where the per capita birth rate equals the per capita death rate the resulting population is at K In the above graph birth rate 2 0019N and death rate 01 0009N By equating the 2 equations and solVing for N we ndKT67857L These curves do not have to be linear See figure 71 of Sinclair 1989 for some more examples Density dependent Rates 12 i U U 5 11 08 5 o l o 06 4 4 a 09 a 04 U U i 08 i 3 07 0392 g 2 Q 06 0 15 3O 45 6O 75 9O Nt For the above graph birth rate 2 0019N15 and death rate 01 0009N5 FW662 Lecture 6 Evidence of density dependence 3 These equations are equal atK 9352678 Likewise the curves do not have to have a single constant K if variation is allowed in the densitydependent functions The idea of densityvague dynamics Strong 1986 results in a cloud of points Both the birth and death functions can exhibit a range of values resulting in a range of values of K illustrated by the shaded area in the following graph Births Demographic Rate V vs K selection The theory of r and K selection was one of the first predictive models for lifehistory evolution and has recently been discussed as a paradigm of ecology by Reznick et al 2002 r and K selection has been displaced by the kinds of demographic models presented in this class However r and K selection still has its place even in these models The shape of the birth and death per capita curve re ects the strategy of the organism on the r and K selection continuum Indirect confrontations scramble or exploitation competition Nutrition affects births deaths CluttonBrock et al 1985 Bartmann et al 1992 McCullough 1979 Lack 1954 Hobbs and Swift 1985 defined quantity of forage as a function of its quality Generally a lot of poor quality forage exists relative to the amount of good quality forage As a result animals must maintain a diet of quality greater than X to be able to maintain a mean concentration in their diet of CONC FW662 Lecture 6 Evidence of density dependence 4 Nutrition Quantity Quality Social Behavior Direct confrontations contest or interference competition Territory size territory provides a link between resources and population processes Space available Plants Harper 1977 General Adaptive Syndrome Calhoun s rats H Selye Chitty s geneticfeedback hypothesis WynneEdwards 1962 1986 group selection advantage accruing to the individual Other resources limiting than food Nest sites Predation and Parasitism More on this later However one example is Competition for shelter space causes densitydependent predation in damself1shes Holbrook and Schmitt 2002 demonstrated that competition for shelter spaces caused more predation in damself1shes These species shelter in branching corals or anemones and when refuge spaces to protect them were filled with their intraspecif1c competitors more mortality from predation was found Andrewartha and Birch 1954 density independence Biological techniques for detecting density dependence Time series of observations with and without perturbation No spatial control Spatial control Quasiexperiments Experimental perturbations Sinclair 1989 FW662 Lecture 6 Evidence of density dependence 5 Statistical techniques for detecting density dependence Tests of density dependence have been developed for a time series of population sizes and as tests of a relationship between birth rates and or death rates and population size Most tests of density dependence are set up with the null hypothesis of density independence and the alternative hypothesis of density dependence Failure to reject the null hypothesis does not constitute evidence of density independence in these cases but only evidence to suggest the test lacked sample size or the experimental variance was too high to reject the null hypothesis In cases where the null hypothesis of density independence is not rejected the investigator should report the con dence interval on the parameter being tested This con dence interval will include the value of the parameter that suggests density independence because the test failed to reject this hypothesis However if this con dence interval is narrow evidence is provided that the true parameter value may not differ much from density independence If the interval is wide evidence is provided that the test lacked power to reject the null hypothesis and hence little information is contained in the data relative to the hypothesis of density dependence No perturbation of population size Procedures for testing density dependence in a time series of population sizes where the population density has not been manipulated have been developed in the entomology literature Bulmer 1975 developed the rst test Pollard et al 1987 extended the procedure and Dennis and Taper 1994 developed the procedure further All of these tests try to detect a return to K carrying capacity That is if the population is density dependent then it should vary around K and not grow inde nitely either direction from K These procedures have not been particularly useful in vertebrate research because the long time series of population sizes needed by these tests have not been available Further all the procedures suffer increased Type I errors when the population sizes are only estimated and hence include sampling error Dennis and Taper 1994 Shenk et al 1998 Considerable controversy has developed over the usefulness of these tests Wolda and Dennis 1993 Holyoak and Lawton 1993 Hanski et al 1993 and Wolda et al 1994 In general their low power makes them ineffective even when the true population size is available Murdoch 1994 Shenk et al 1998 Manipulation of population size FW662 Lecture 6 Evidence of density dependence Number of Individuals Nt When the population size has been manipulated to a level belowK and observed as it increases or alternatively just manipulated to different levels and the amount of growth to the next year observed a much more powerful approach to detecting density dependence is provided Examples include the introduction of a species to an area previously uninhabited or intensive harvest As an example consider the growth of a ringnecked pheasant Phaisianus colchz39cus population introduced to an island off the coast of the state of Washington Initially in 1937 2 cocks and 6 hens were introduced The data on population sizes from 193742 are 8 30 81 282 7051325 Einarson Pheasants 2000 1500 1000 500 Year t Lack 1954 commented The gures suggest that the increase was slowing down and was about to cease but at this point the island was occupied by the military and many of the birds were shot The dashed line in the gure represents densityindependent population growth ie the model N Noe quot The solid line represents densitydependent population growth ie the differential equation model The densitydependent model provides the best fit of the data but then it should It has 2 parameters compared to only 1 parameter for the density independent model Is the improvement of the more complex model statistically significant and hence is densitydependence supported by FW662 Lecture 6 Evidence of density dependence 7 these data Was Lack correct in suggesting that growth had indeed slowed down To answer these questions we can either take a hypothesis testing paradigm and construct a statistical test of the null hypothesis that the 2 models fit equally well or use informationtheoretic methods and consider the weight of the evidence for each model First compute the sum of squared deviations for each model or the SSE for each model Model S SE Dependence 142 79 Independence 5960002 Clearly the difference in SSE suggests a difference To test the null hypothesis of no difference between the 2 models construct an F test as follows SSE I 7 SSE D F df de dfz dev dfn SSED de where SSE D is the sum of squared errors for the densitydependent model SSE I is the sum of squared errors for the densityindependent model and de and de are the respective degrees of freedom of the 2 models The F statistics equals 12492 with 1 and 3 degrees of freedom P lt 0001 Thus we conclude that Lack was correct Some readers may question the use of normal theory to develop this test Note that this assumption can be changed by fitting a model with multiplicative and hence lognormally distributed errors or by randomization or permutation tests For multiplicative errors the F test is 38155 P 0009 still supporting the conclusion that the population was exhibiting density dependence and that Lack was correct However the attempt here is to demonstrate the technique not provide a full treatment of the statistical analyses For an informationtheoretic approach we compute the AICc and Akaike weights for each model Bumham and Anderson 2002 The AICc and Akaike weight for each model can be computed directly from the SSE for the 2 models Model S SE AIC Weight FW662 Lecture 6 Evidence of density dependence 8 Dependence 142 79 46 76 0994 Independence 5960002 5693 0006 As with the hypothesis test paradigm the hypothesis of density dependence is strongly supported Given that the population has been manipulated the birth and death rates may also be observed and tests of density dependence developed for changes in these rates as a function of population size Such an approach was used by Bartmann et al 1992 to test for changes in overwinter survival of mule deer fawns In their experiment fawn survival was estimated by radio tracking animals and density was known because a set number of animals were stocked in each enclosure Often however attempts are made to estimate per capita recruitment to the next time step from a sequence of observed population estimates As discussed next such an approach must be implemented carefully Problem of induced correlation NM NN regressed against N NIHNZ regressed against NZ even worse Eberhardt 1970 demonstrated that a correlation of r07 is expected In the following graph Y andX are random normal variables with mean 10 and standard deviation 1 As the first graph demonstrates the 2 variables were independent ie no relationship exists szX 14 135 13 125 gt 12 115 11 105 As the following graph shows the regression of YX versus X is significant and the correlation is close to the 07 that Eberhardt 1970 suggested it should be That is 0475 2 0689 FW662 Lecture 6 Evidence of density dependence 9 YIX vs X 476 P 09 08 Procedures to test for a relationship between recruitment including per capita birth or death rates and population size have been extensively used to test for density dependence Tanner 1966 McCullough 1979 The approach is to regress population growth rate against population size If population growth is density independent the expected slope of the regression is 0 If density dependent growth is occurring the slope of the regression should be negative with a negative correlation However as rst pointed out by Eberhardt 1970 population growth rate recruitment must be estimated independently of population size When population growth rate is estimated from the time series of population sizes as and R is regressed against N t a correlation is induced because N occurs on both sides of the regression Eberhardt 1970 pointed out that correlations of about 07 are expected for sequences of random numbers when tested with the regression procedure used by Tanner 1966 Review of evidence by major animal groups good review by Sinclair 1989 183911 Century Humans Matessi and Menozzi 1979 Birth rates did not depend on population size but did increase from mountains to hills to plains whereas death rates increase with population size and also increase from mountains to hills to plains Consequently population growth rates were density dependent with spatial variation Large Herbivores 7 nutrition Fowler 1987 Eberhardt 2002 suggested a FW662 Lecture 6 Evidence of density dependence 10 paradigm for population regulation of longlived vertebrates A sequence of changes in vital rates observed as populations approach maximal levels has been used as the basis for a paradigm for population analysis Past work indicates that early survival survival of young animals decreases first followed by lower reproductive rates ultimately adult female survival may decrease Sensitivity of population growth rates as measured by partial derivatives of Leslie matrix models follow the same sequence suggesting that population regulation follows this sequence and implying some evolutionary significance in the sequence Large Camivores 7 social territory size Raptors Newton 1991 Stability of breeding population Existence of surplus adult Reestablishment of populations to same level prior to removal Regular spacing of nests where nest sites are not limited Small Mammals Meadow voles 7 strong densitydependent regulation via birth rates Ostfeld et al 1993 They suggest cycles not caused by lagged effects of resource exploitation Grouse Red grouse 7 cycles not stopped based on maternal nutrition a version of Chitty s genetic hypothesis hostparasite caecal threadworm or predatorprey relationships Moss et al 1996 They suggest the changing age structure in the population as one explanation Insects Why is the evidence so weak Sinclair 1989 Discussion Bring to class an example of a good test of densitydependence and be prepared to discuss what critical information is lacking to produce a valid population model that incorporates densitydependence Literature Cited Andrewartha H G and L C Birch 1954 The distribution and abundance of animals University Chicago Press Chicago Illinois USA 782 pp Bartmann R M G C White and L H Carpenter 1992 Compensatory mortality in a Colorado mule deer population Wildlife Monograph 121 139 Bulmer M G 1975 The statistical analysis of density dependence Biometrica 31901911 Bumham K P and D R Anderson 2002 Model selection and multimodel inference a practical information theoretic approach Second Edition Springer Verlag New York FW662 Lecture 6 Evidence of density dependence 11 New York USA Christian J J and D E Davis 1964 Endocrines behavior and population Science 1461550 1560 CluttonBrock T H M Major and F E Guinness 1985 Population regulation in male and female red deer Journal of Animal Ecology 54831846 Dennis B and M Taper 1994 Density dependence in time series observations of natural populations detecting stability in stochastic systems Ecological Monograph 64205224 Eberhardt L L 2002 A paradigm for population analysis of longlived vertebrates Ecology 8328412854 Eberhardt L L 1970 Correlation regression and density dependence Ecology 51306310 Fowler C R 1987 Overview of density dependence in populations of large mammals Pages 401441 in H H Genownys ed Current Mammalogy volume 1 Plenum New York New York USA Hanski I I Woiwod and J Perry 1993 Density dependence population persistence and largely futile arguments Oecologia 95595598 Harper J L 1977 Population biology of plants Academic Press New York New York USA 892 pp Hassell M P 1986 Detecting density dependence Trends in Ecology and Evolution 19093 Hixon M A S W Pacala and S A Sandin 2002 Population regulation historical context and contemporary challenges of open vs closed systems Ecology 83 14901508 Hobbs N T and D M Swift 1985 Estimates of habitat carrying capacity incorporating explicit nutritional constraints Journal of Wildlife Management 49814822 Holbrook S J and R J Schmitt 2002 Competition for shelter space causes density dependent predation mortality in damselflshes Ecology 8328552868 Holyoak M and J H Lawton Comment arising from a paper by Wolda and Dennis using and interpreting the results of tests for density dependence Oecologia 95592594 Lack D L 1954 The natural regulation of animal numbers Clarendon Press Oxford United Kingdom 343 pp FW662 Lecture 6 Evidence of density dependence 12 Matessi C and P Menozzi 1979 Environment population size and vital statistics an analysis of demographic data from 18 11 century villages in the province of Reggio Emilia Italy Ecology 60486493 McCullough D R 1979 The George Reserve deer herd University Michigan Press Ann Arbor Michigan USA 271 pp Moss R A Watson and R Parr 1996 Experimental prevention of a population cycle in red grouse Ecology 77 15121530 Murdoch W W 1994 Population regulation in theory and practice Ecology 75271287 Newton 1 1991 Population limitation in birds of prey a comparative approach Pages 321 in C M Perrins JD Lebreton and G J M Hirons eds Bird Population Studies Oxford New York New York USA Ostfeld R S C D Canham and S R Pugh 1993 Intrinsic densitydependent regulation of vole populations Nature 366259261 Pollard E K H Lakhani and P Rothery 1987 The detection of density dependence from a series of annual censuses Ecology 6820462055 Resnick D M J Bryant and F Bashey 2002 r and K selection revisited the role of population regulation in lifehistory evolution Ecology 83 15091520 Shenk T M G C White and K P Bumham 1998 Samplingvariance effects on detecting density dependence from temporal trends in natural populations Ecological Monographs 86445463 Sinclair A R E 1989 Population regulation in animals Pages 197241 in J M Cherrett ed Ecological concepts Blackwell Scienti c Publishers Oxford United Kingdom Strong D R 1986 Densityvague population change Trends in Ecology and Evolution 139 42 Tanner J T 1966 Effects of population density on growth rates of animal populations Ecology 47733745 Wiens J A 1977 On competition and variable environments American Scientist 65590597 Wolda H and B Dennis 1993 Density dependence tests are they Oecologia 95581591 Wolda H B Dennis and M Taper 1993 Density dependence tests and largely futile FW662 Lecture 6 Evidence of density dependence 13 comments Answers to Holyoak and Lawton 1993 and Hanski Woiwod and Perry 1993 Oecologia 98229234 WynneEdwards V C 1962 Animal dispersion in relation to social behavior Oliver and Boyd Limited Edinburgh Scotland WynneEdwards V C 1986 Evolution through group selection Blackwell Scienti c Palto Alto California USA 386 pp FW662 Lecture 5 Age structured models 1 Lecture 5 Age and stage structured models Leslie Le iovitch Models Reading Gotelli 2001 A Primer of Ecology Chapter 3 pages 4980 Noon B R and J R Sauer 1992 Population models for passerine birds structure parameterization and analysis Pages 441464 in D R McCullough and R H Barrett eds Wildlife 2001 Populations Elsevier Applied Science New York New York USA Optional Lande R 1991 Population dynamics and extinction in heterogeneous environments the Northern Spotted Owl Pages 566580 in C M Perrins J D Lebreton and G J M Hirons eds Bird Population Studies Oxford New York New York USA De ne age structure dynamics in terms of difference equations N is population at time tof age class 1 Time tis start of biological year or the time of reproduction With k age classes the maximum age an animal can attain is k so that the survival rate of animals k years old is 0 Only females are considered in the following example The modeler must define the anniversary date of the population census For the following equations the animal is incremented in age ie mortality takes place then reproduces The population census is E the birthpulse Define j as the number of young produced by animals of age 1 and S as survival rate of animals of age 1 to end of the year Animals just born are in the N0 age class Survival to next age class N111Na1 X 30 N211N11 X S N311 N21 X 32 Nicm Nkiu X 5171 Reproduction Na11N111Xf1 N211 sz th1gtlt r However we do not want the equations to refer to the t 1 populations but rather the tpopulations Therefore we must substitute the survival equations into the reproduction equation 71711701X So sz Nu X31X N1ru X3171 ka FW662 Lecture 5 Age structured models 2 Now construct the Leslie 1945 1948 matrix also known as a projection matrix or transition matrix based on the above difference equations MJ L X A 307i sz Smirk 0 N 0 31 0 0 x In the following equations reproduction takes place then mortality The census of the population is before the birthpulse The definitions of the and s1 remain the same as the after birthpulse model However the interpretation of the M changes slightly because now the population sizes are just prior to reproduction Hence there are no newly born animals in this model corresponding to the N0 of the after birthpulse model Rather NZ corresponds to animals 2 days younger than in the previous model That is N1 in the after birthpulse model was the population of animals aged 1 year a day Now in the before birthpulse model N1 is the population of animals aged 1 year a day ie just before they reach their first birthday As a result the N0 age class is no longer being modeled directly although the survival rate of animals from birth to 1 year of age so is still in the model Survival to next age class N2z1N1z X S N3z1 N2z X 32 Nam NHz X Sk Reproduction N1z1N12X X 30 N22 sz X 30 Nm ka X so Construct the Leslie matrix N1LXM FW662 Lecture 5 Age structured models 3 130 fzso f350 kaO 1 s1 0 0 0 2 H17 0 s2 0 0 x 3 0 0 0 sk 0 kt Carefully note the differences in the 2 Leslie matrices In the after birthpulse matrix the top row contains the survival rates of the reproducing animals In the before birth pulse matrix the top row contains the survival rate of new born animals to 1 year of age The most common mistake with application of the Leslie notation is that the presenter confuses order of the birth and death process Noon and Saur 1992 discuss how to configure the matrix for use of estimates of survival and reproduction from a population Benefits of Leslie matrix formulation A eigenvalue ofL rate ofpopulation growth ie NM ANI Equilibrium age ratios are eigenvector of L ie ratio of N1 N2 is the same as the ratio of the first 2 values in the eigenvector The stable age distribution is the ratio of the various age classes to one another These ratios are stable in a Leslie matrix projection regardless of the value of A For A l the age distribution is termed stationary because the population is stationary Ease of presentation if you know matrix algebra Construction of a spreadsheet model from this formulation Provides a simple available alternative to model a population Ratio of NMNt after a few generations provides A Ratio of age classes is equivalent to eigenvector after a few generations Interpretation of age ratios measured in a population Because the projection matrix generates a stable age distribution after effects of initial population size are depreciated age ratios tell you nothing about increase or decrease of a population quotTo sum up age ratios cannot be interpreted without a knowledge of rate of increase and if we have an estimate of this rate we do not need age ratiosquot Caughley 1974 Example of eigenvalue calculation 030 fISO 310 FW662 Lecture 5 Age structured models Eigenvalues are values of A that solve 0 detL 7 M where I is the identity matrix 030 fISO s1 0 030 7 A fISO s1 7 0 7Afoso A2 7f1sos1 A0 0 0 det 0 det 0 A2 7 0330 7 flsosl Remember that for the polynomial ax2 bx c the roots are 7b i Vbz 7 4610 2a Thus the roots of the characteristic equation are 2 2 foso l Vfoso 4113031 2 and 2 2 foso Vfoso 4113031 2 with the rst root the largest value or dominant eigenvalue Typically the analytical eigenvalues are seldom used and only the FW662 Lecture 5 Age structured models numerical value provided Computing the eigenvalues of a matrix is a standard numerical analysis problem The basic Leslie Matrix formulation is limited because only densityindependent population growth with just births and deaths is modeled The following examples all with the census before the birthpulse are some approaches to extend the basic formulation to incorporate additional population processes Exponential growth is what the above examples portray ie all rates are density independent The matrix can be modified to make and 3 functions of the total population size N Another possibility is to make some age specific parameters densitydependent only on the size of a specific age class In both cases if these functions are linear relationships then logistic population growth results In the following example assume that N t is the sum of the population sizes for all the age classes Then replace survival for the first age class with the function 30Nz Bo T 3th so that survival of this age class is now densitydependent The matrix would look like the following 050 7 l3th f1Bo 7 l3th fzajo 7 l3th fkl30 7 l3th 0 s1 0 0 0 1 0 32 0 0 x 2 0 0 0 sk 0 k I Only densitydependence in this one age class is needed to produce logistic growth For discussion what is the value of A for t a co Would densitydependence in just one of the also result in density dependent population growth Hint if you set a parameter to zero and A remains greater than 1 what is the impact on population growth of making the parameter densitydependent Immigration is precluded ie the population is closed to immigrants However an extension to the Leslie matrix is to include immigration with the following technique where is the number of immigrates of age 139 5 FW662 Lecture 5 Age structured models 030 fiso fzso kaO 0 10 31 0 0 0 1 1 t1 0 32 0 0 X 2 12 0 0 0 sk 0 kt Ikr Emigration can be considered part of the death process but this is rather ad hoc A more sophisticated approach is to treat the survival entries as survival minus emigration as shown in the following example where the el values are the emigration rate for animals of age 139 030 7 80 fl30 7 80 f230 7 80 fkso N0 S1 el 0 0 0 N1 t1 0 S2 ez 0 0 X N2 0 0 0 Sk 8k 0 th Harvest can be treated either as an absolute loss from the population such as 030 fISO fzso kaO 0 s1 0 0 0 1 H17 0 s2 0 0 x 2 7 0 0 0 sk 0 kt where the H I values are the number of animals harvested of each age class Another approach is more like the emigration example above where a harvest rate hi is applied to each age class such as the following 050 7 h0 f1SO 7 ho fz30 7 ho kaO No s1 7 hl 0 0 0 N1 1 1 0 s2 7 hz 0 0 x N2 FW662 Lecture 5 Age structured models 7 Environmental stochasticity can be incorporated into a Leslie matrix by making the parameters random variables Also demographic stochasticity can be included by applying the rates are random processes Thus instead of multiplying by the survival rate of S1 a binomial process is applied to the update from N1 to N2 A similar strategy can be used for fecundity rates Another common extension is to allow the last age class to continue to survive and reproduce at constant rates This is accomplished by setting the survival rate for the maximum age in population in the lowerright comer of the matrix Le ltovitch 1965 Usher 1966 Hence the label quotcomer trickquot Sofo Slfl 322 3323 342 0 so 0 0 0 0 1 it 1 0 SI 0 0 0 x 2 0 0 32 0 0 3 0 0 0 s3 s4 4 In this matrix the maximum age animals do not suffer 0 survival but persist with probability s4 The implication is that there is no senescence in the population ie that older animals continue to perform the same in terms of reproduction and survival Depending on the proportion of the population in this age class the assumption of no senescence may be invalid For example Ericsson et a1 2001 demonstrated that female moose Alces alces showed senescence in reproduction ie litter size from about 12 yr of age Further evidence of senescence was a decrease in parental care during summer expressed as increased offspring mortality with the mother s age A compensating mechanism for senescence may come with heavy harvest in that few animals achieve an age whereby senescence is exhibited Instead of treating the population values as ages these values can be treated as stages of a population life cycle Stagestructured population models were developed by Usher 1966 1969 with stage meaning the stage of development ie insect instars N now represents the number of animals of stage 139 not age 139 Then animals can remain in the same stage or be advanced to the next stage or even advanced more than one stage The following matrix is an example from a stagestructured population for loggerhead sea turtles Caretta caretta from Crouse et a1 1987 Seven life stages were considered 1 eggs and hatchlings lt1 year 2 small juveniles 17 years 3 large juveniles 815 years 4 subadults 1621 years 5 novice breeders 22 years 6 firstyear FW662 Lecture 5 Age structured models emigrants 23 years and 7 mature breeders gt23 years 0 0 0 0 127 4 80 06747 07370 0 0 0 0 0 0 00486 06611 0 0 0 0 0 0 00147 06907 0 0 0 0 0 0 00518 0 0 0 0 0 0 0 08091 0 0 0 0 0 0 0 08091 08089 Reproduction is from the last 3 stages only with fecundity rates being 127 4 and 80 each year respectively At stage 2 for example the probability of surviving and remaining in the stage from one year to the next is 07370 whereas the probability of advancing to stage 3 is 00486 The annual survival rate of stage 2 animals is 07370 00486 07856 so 1 07856 is the annual mortality rate For this matrix A 09450 so the population is expected to decline A modi cation of this model was used by Crowder et a1 1994 to evaluate turtle excluder devices on trawl sheries of the southeastern US Variance of A as a function of the variance of the entries in the projection matrix is given by this Taylor series approximation Delta method 2 6A 6A i j 611 611 where variance of A is expressed as the partial of A with respect to parameters 111 and 11 entries in the matrix The 111 and 11 are not ce11 values but rather correspond to the basic parameters used to create the cell values For example 111 might correspond to f1 112 to f2 113 to SI etc Covariance of 111 and 11 is 0U The sensitivity of A is the partial of A with respect to parameter Tri 01 and 0 may represent just process variation just sampling variation or a combination of both For determining minimum viable population size you want Oi to just represent process variation To estimate the sampling variation of A you want 01 to just represent sampling variation Lande 1991 uses this procedure with northern spotted owls also discussed by Caswell 2001 Gross 2002 describes how to allocate effort in data collection to minimize the sampling variance of A Sensitivity analysis for the mathematically challenged To determine relative sensitivity of A with respect to SI FW662 Lecture 5 Age structured models 9 1 compute A with the given value of SI 2 compute A A with the value 51 A where A is some small value compared to S 1 3 compute the sensitivity of A with respect to SI as E M AS1 A This approach is equivalent to computing numerical partial derivatives and as A approaches zero the resulting sensitivity value approaches This procedure 3 1 is relatively easy to perform in a spreadsheet In general projection matrix models demonstrate that the adult survival rate particularly the parameters in the lower right corner is the most sensitive parameter whereas reproductive rates and juvenile survival have the same sensitivity with A being less sensitive to them A common misconception is that the most sensitive parameters are the most important for the persistence of the population As we will see later the variation of a parameter across time space and individuals affects persistence and is unrelated to sensitivity However even though the concepts of parameter sensitivity and process variance are unrelated most populations do show a relationship This is because parameters to which A is highly sensitive probably do not have a large process variance because if this is the case the population would likely go extinct A nifty use of the Leslie matrix is demonstrated by McGraw and Caswell 1996 where they define individual fitness as the dominant eigenvalue of an individual s Leslie matrix Thus reproduction for the individual is the actual number of offspring produced and the survival rate for the individual is 1 until the animal dies when it is zero The dominant eigenvalue of the matrix is then the animal s fitness A general reference on projection matrices is Manly 1990 A mathematical treatment is given in Caswell 1989 2001 Laboratory Exercise 4 Quattro spreadsheet with agestructured population Literature Cited Caswell H 1989 Matrix population models Sinauer Sunderland Massachusetts USA 28 pp FW662 Lecture 5 Age structured models 10 Caswell H 2001 Matrix population models 2nd Edition Sinauer Sunderland Massachusetts USA 722 pp Caughley G 1974 Interpretation of age ratios Journal of Wildlife Management 38557562 Crouse D T L B Crowder and H Caswell 1987 A stagebased population model for loggerhead sea turtles and implications for conservation Ecology 68 14121423 Crowder L B D T Crouse S S Heppel and T H Martin 1994 Predicting the impact of turtle excluder devices on loggerhead sea turtle populations Ecological Applications 4437445 Ericsson G K Wallin J P Ball and M Broberg 2001 Agerelated reproductive effort and senescence in freeranging mooseAlces alces Ecology 8216131620 Gross K 2002 Efficient data collection for estimating growth rates of structured populations Ecology 831762 1767 Lande R 1991 Population dynamics and extinction in heterogeneous environments the Northern Spotted Owl Pages 566580 in C M Perrins JD Lebreton and G J M Hirons eds Bird Population Studies Oxford United Kingdom Le ltovitch L P 1965 The study of population growth in organisms grouped by stages Biometrics 21118 Leslie P H 1945 On the use of matrices in certain population mathematics Biometrika 33 183212 Leslie P H 1948 Some further notes on the use of matrices in population mathematics Biometrika 35213245 Manly B F J 1990 Stagestructured population sampling analysis and simulation Chapman and Hall London United Kingdom 187 pp Noon B R and J R Sauer 1992 Population models for passerine birds structure parameterization and analysis Pages 441464 In D R McCullough and R H Barrett eds Wildlife 2001 Populations Elsevier Applied Science New York New York USA Usher M B 1966 A matrix approach to the management of renewable resources with special reference to selection forests Journal of Applied Ecology 3355367 Usher M B 1969 A matrix model for forest management Biometrics 25309315 FW662 Lecture 11 Competition 1 Lecture 11 Competition Reading Gotelli 2001 A Primer of Ecology Chapter 5 pages 99 124 Renshaw 1991 Chapter 5 Competition processes Pages 128165 Optional Schoener T W 1983 Field experiments on interspecific competition American Naturalist 122240285 Competition The negative effects which one organism has upon another by consuming or controlling access to a resource that is limited in availability Keddy 19892 The ow of energy through trophic levels follows the second law of thermodynamics There is less and less energy available to each successive tropic level resulting in the classic trophic level pyramid However this simple decline is not evidence that energy is lacking or limiting Since competition often puts a premium on efficiency this assumption implies a division of labor among specialists It is the ultimate reason we have so many species MacArthur 1972 Competitive Exclusion Principle was proposed by Gause 1934 as a Law although plant ecologists held this view prior to Gause Law and Watkinson 1989 Two species cannot coexist on a single limiting resource if there is no differentiation between the realized niches of 2 competing species or if such differentiation is precluded by the limitations of the habitat then one species will eliminate or exclude the other Begon and Mortimer 198678 Intraspecific competition leads to stable populations in that density dependence of birth and death rates results in the population approaching a carrying capacity because eventually birth rate equals death rate However these examples may be evidence for competition because 1 The results are correlational and some other factor may cause the decline in the birth and survival rates such as predation 2 Keddy 19892830 discusses some other more obscure reasons How does competition occur Park 1962 exploitative competition use of resources deprives others interference competition direct harm to others by physical or chemical means Schoener 1983 Consumptive competition use of same renewable resource Preemptive competition occupation of open space Overgrowth competition one individual grows over another depriving the second of resources Chemical competition production of toxin allelopathy Territorial competition defense of space Encounter competition transient interaction over a resource ie theft of food FW662 Lecture 11 Competition 2 LotkaVolterra competition equations provide a model of interspeci c competition The Lotka Volterra models are so popular in the study of ecology that the study of the equations themselves is frequently recognized as ecological research Simberloff 1983a Fagerstrom 1987 in Keddy 198950 These are the same LotkaVolterra equations of the predatorprey models but the interpretation of the coef cients is somewhat different alN K e a N e a 1 rlNl 1 11 1 12N2 dt K1 rZN K2 7 a21N1 azzNz alt 2 K 2 By de nition each species competition coef cient upon itself is 1 ie a11 1 a22 1 The coef cient a12 is the competition coef cient of N2 on N1 and a21 is the competition coef cient of N1 on N2 If a12 equals zero then the rst equation results in logistic growth for species 1 ie species 2 has no impact on its population growth Likewise if a21 equals zero then the second equation results in logistic growth for species 2 with no impact of species 1 on its growth Deterministic Competition 0 20 40 60 80 100 120 Time Each population grows with logistic function in absence of the other Further the competitive interactions modeled by the competition coef cients a12 and a21 between the 2 species are independent of the densities which is unlikely Ayala et a1 1973 SmithGill and Gill 1978 and Law and Watkinson 1987 We cannot solve the above equations analytically except for special cases Hence we use difference equations to approximate the solutions numerically Also we can study equilibrium conditions by setting the equations equal to zero ie no change is taking place either population We quickly see that if either r 0 0r r2 0 then the population growth rate goes to zero More interesting is to set K17N17a12Nz0 FW662 Lecture 11 Competition 3 N2 This equation predicts the nal population size of N2 as a linear function of the nal population size of N2 The intercept of the line is K 161 12 and the slope is 16112 The following graph results KlalZ Kl N1 A similar graph results for N 1 in terms of N2 From these graphs and some algebra we see that a gt KKZ and an gt KK results in one species winning depending on initial conditions a lt KK2 and an gt KK results in species 1 winning a gt KKZ and 6121 lt KK results in species 2 winning and a lt KKZ and 6121 lt KK results in both species coexisting The competitive exclusion principle implies that one of the 2nd or 3rd conditions listed above is true and hence one species will always win However the number of extenuating circumstances make the competitive exclusion law useless Further examples of competitive exclusion are lost because we never see the species that lost Ghosts of competition past Pascual and Kareiva 1996 have t the LotkaVolterra equations to observed data to evaluate between species FW662 Lecture 11 Competition 4 These models have all the same limitations as the logistic equation There is no room in these equations for Variation in competition through time Spatial variation Environmental variation Wiens 1977 suggested competition is likely to be important only when the environment is stable which is the only time these equations can apply Environmental variation in the form of temporal changes precludes the logistic assumptions ie constant K 1 and K 2 Spatial variation means that the coefficients a 12 and 6121 would not be constant but random variables because refuges would be created or in general competition would not be constant over space As a result all the analyses that result in coexistence probably is an artifact of this model Emlen l9154 suggests temporal change including the interactive changes in predator populations and spatial structuring of the environment seem to vitiate the hypothesis Stochastic equations are provided by Renshaw 1991 that incorporate only demographic stochasticity Stochastic Competition N 01 O I Populations H O This model will not result in coexistence of 2 species Cause and effect relationships Law and Watkinson 1989 Asymmetry in manipulation studies Experimental studies are required to demonstrate that one species is causing another to decline Several approaches are possible including adding members of one of the populations removing members of one of the populations substituting members of one population with another and all of the above together response surface FW662 Lecture 11 Competition 5 design Additive experiments Maybell elk study Hobbs et al 1996ab The hypothesis was that elk grazing on sagebrush rangelands during the winter caused competition for cattle during the following spring The 2 species are not present on the area at the same time Ranchers want game damage payments for elk grazing As a result a political carrying capacity for elk is established to minimize game damage payments The experiment involved a large circular area divided up into 12 pie wedges In these 12 areas 4 levels of elk grazing were introduced with 3 replicates of each Cattle were then stocked in the 12 areas The weight of the cow was taken when stocked and when removed and the same weights of her calf In addition calf weights at the end of the summer after high country grazing were also taken Calf weights were shown to decline in response to elk density Possible explanations are forage parasites or disease Is this study a demonstration of competition Removal experiments Connell 1961 Competitive exclusion is accomplished by direct physical interaction Abstract Adults of Chthamalus stellatus occur in the marine intertidal in a zone above that of another barnacle Balanus balanoides Young Chthamalus settle in the Balanus zone but evidently seldom survive since few adults are found there The survival of Chthamalus which had settled at various levels in the Balanus zone was followed for a year by successive censuses of mapped individuals Some Chthamalus were kept free of contact with Balanus These survived very well at all intertidal levels indicating that increased time of submergence was not the factor responsible for elimination of C hthamalus at low shore levels Comparison of the survival of unprotected populations with others protected by enclosure in cages from predation by the snail Thais lapillus showed that Thais was not greatly affecting the survival of Chthamalus Comparison of the survival of undisturbed populations of Chthamalus with those kept free of contact with Balanus indicated that Balanus could cause great mortality of Chthamalus Balanus settled in greater population densities and grew faster than C hthamalus Direct observations at each census showed that Balanus smothered undercut or crushed the Chthamalus the greatest mortality of Chthamalus occurred during the seasons of most rapid growth of Balanus Even older Chthamalus transplanted to low levels were killed by Balanus in this way Predation by Thais tended to decrease the severity of this interspecifrc competition Survivors of Chthamalus after a year of crowding by Balanus were smaller than uncrowded ones Since smaller barnacles produce fewer offspring FW662 Lecture 11 Competition 6 competition tended to reduce reproductive efficiency in addition to increasing mortality Mortality as a result of intraspecies competition for space between individuals of Chthamalus was only rarely observed The evidence of this and other studies indicates that the lower limit of distribution of intertidal organisms is mainly determined by the action of biotic factors such as competition for space or predation The upper limit is probably more often set by physical factors Response surface experiments Inouye 2001 Inouye 2001 suggests that additive or substitution designs limit the usefulness of experiments in ecology because the inferences are limited from the resulting data He suggests response surface designs where the densities of the 2 competing species are varied independently remedy the issue He found that response surface designs more accurately estimated the parameter values in simulations and thus provide a stronger connection between empirical and theoretical approaches than traditional experimental designs Goals of competition research In the future we want to be able to predict introduction into an existing community Will barbery sheep compete with mule deer Are elk expanding in Colorado at the expense of mule deer size of conservation areas required to allow 2 competing species to coexist Literature Cited Ayala F J M E Gilpin and J G Ehrenfeld 1973 Competition between species theoretical models and experimental tests Theoretical Population Biology 4331356 Connell J H 1961 The in uence of interspecific competition and other factors on the distribution of the bamacle Chthamalus stellatus Ecology 42710723 Gause G F 1934 The struggle for existence Hafner New York New York USA pp Hobbs N T D L Baker G D Bear and D C Bowden 1996a Ungulate grazing in sagebrush grassland mechanisms of resource competition Ecological Applications 6200217 Hobbs N T D L Baker G D Bear and D C Bowden 1996b Ungulate grazing in sagebrush grassland effects of resource competition on secondary production Ecological Applications 6218227 Inouye B D 2001 Response surface experimental designs for investigating interspecific competition Ecology 8226962706 FW662 Lecture 11 Competition 7 Keddy P A 1989 Competition Chapman and Hall London United Kingdom 202 pp Law R and A R Watkinson 1987 Responsesurface analysis of twospecies competition an experiment on Phleum arenarium and Vulpz39afasciculata Journal of Ecology 75871 Law R and A R Watkinson 1989 Competition Pages 243284 in J M Cherrett ed Ecological concepts Blackwell scienti c Publ Oxford United Kingdom Park T 1962 Beetles competition and populations Science 13813691375 Pascual M A and P Kareiva 1996 Predicting the outcome of competition using experimental data maximum likelihood and Bayesian approaches Ecology 77337349 Schoener T W 1983 Field experiments on interspeci c competition American Naturalist 122240285 SmithGill S J and D E GIllinois USA 1978 CurVilinearities in the competition equations an experiment with ranid tadpoles American Naturalist 112557570 Weins J A 1977 On competition and variable environments American Scientist 65590597 FW662 Lecture 8 Spatially structured populations 1 Lecture 8 Spatially structured populations Reading Gotelli 2001 A Primer of Ecology Chapter 4 pages 8197 Hanski I 1996 Metapopulation ecology Pages 1343 in Rhodes 0 E Jr R K Chesser and M H Smith eds Population dynamics in ecological space and time University Chicago Press Chicago Illinois USA Pulliam H R 1996 Sources and sinks empirical evidence and population consequences Pages 4569 in Rhodes 0 E Jr R K Chesser and M H Smith eds Population dynamics in ecological space and time University Chicago Press Chicago Illinois USA Optional Gilpin M E 1987 Spatial structure and population vulnerability Pages 125139 in M E Soule ed Viable Populations for Conservation Cambridge University Press New York New York USA 189 pp Kareiva P 1990 Population dynamics in spatially complex environments theory and data Pages 5368 in Population Regulation and Dynamics M P Hassell and R M May eds The Royal Society London United Kingdom Traditional View of Populations Panmictic no selective mating Frequency of genes in the population follows Hardy Weinberg equilibrium Equilibrium population reaches a carrying capacity Extinction dependent on N Homogeneous environmenthabitat no spatial heterogeneity 39 J39 39J 39 equalno39 quotyof39 quot 39 We have relaxed some of these assumptions in previous models stochastic models age structured models However basically previous models have been quottraditionalquot Difficulty of incorporating spatial extension into population models Difficult to keep track of locations of organisms in field studies Computational problems of dealing with space in a theoretical manner are formidable requiring computer models Spatial patterns in populations Traditional approach is a diffusion equation Start with a model of local population dynamics dN N M dt where N is the number of individuals of the species at time t and fN is the per capita growth rate of the species If we now add movement based on a diffusion FW662 Lecture 8 Spatially structured populations 2 equation Hastings 1990 N depends on both time and space x and N is now a density on space x for the number of individuals A typical form Hastings 1990 is 6Nx t 6D 6Nx t N N 6t 6x where D which may depend on x is a measure of the speed of movement of individuals CentralPeripheral Wright 1943 SourceSink Gill 1978 Lidicker 1975 Pulliam 1996 Pulliam 1988 argued that active dispersal from source habitats can maintain large sink populations and that such dispersal may be evolutionarily stable The key idea is that if the source is saturated an individual without a territory in the source is better off to migrate to the sink because some chance of breeding is better than none Van Home 1983 suggested such a system was operating with populations of Peromyscus maniculatus in spruce and hemlock stands of different seral stages in southeast Alaska The implication of this model is that density is not necessarily a good predictor of habitat quality in terms of maintaining the species Metapopulation Levins 1970 denBoor 1968 Traditional CentralPeripheral Metapopulation SourceSink O O 0746 3 O O Other examples see Fig 72 of Gilpin 1987 Habitat occupancy rate FW662 Lecture 8 Spatially structured populations a CentralPeripheral 3 Island Biogeogmphy a Metapopulation High Number of Key features of spatial heterogeneity models Localized extinction each patch has some probability of extinction within the patch The patch is recolonized by immigration dispersal from other patches The percent of patches occupied is a key variable Extinction and recolonization dynamics The model behavior is critically in uenced by the extinction and recolonization rates If the extinction rate is gtgt re colonization rate the population goes extinct If the recolonization rate gtgt extinction rate the model is really just a single population and the spatial heterogeneity is covered up As a result of the almost complete lack of ecological data to support these models almost all the evidence comes from mathematical models These models are hypotheses to be disproved Currently the theory hypothesis in vogue is a metapopulation Features of r r 39 im quotr r 39 quot ofr r 39 quot quot Hanski 1996 Weins 1996 Many local populations gt2 10 Each local population has a given extinction probability Each local population has traditional dynamics hence you don t have to forget all you have learned so far High probability of extinction generates quotwinkingquot no probability of extinction generates quot xedquot populations Dispersal is responsible for recolonization of vacant sites Spatial subdivisions enhance stability of total population quotspreading of riskquot denBoer 1968 quotthe risk of wide uctuations in animal numbers is spread unequally over a number of subpopulationsquot An exception to idea are catastrophes large enough in the spatial scale to blanket the entire metapopulation FW662 Lecture 8 Spatially structured populations Metapopulation stability requires Asynchrony or otherwise you have just a single large population Densitydependence or a single patch eventually goes to in nity Large number of units Central Limit Theorem to stabilize the system Dispersal must not be high enough to synchronize local populations or otherwise the patches all operate in synchrony and the result is a single large population Dispersal must be sufficient to offset extinction of a patch or else eventually all the patches will go extinct Gene flow must be great enough to prevent localized selection from generating new species Conclusions about metapopulations have basically come from models data are lacking for real systems An example model Levins 1969 d p mp1pep dt where p proportion of patches occupied e rate of local extinction with each patch having the same extinction probability and m rate of colonization m is a function of dispersal and is the same for each patch The spatial configuration of the patches is ignored in this model The equilibrium condition is mp1 p ep 0 sop 1 em The model also assumes nothing about within patch population dynamics only that a patch goes extinct and is recolonized At any time a patch is either atK carrying capacity or extinct 0 This differential equation is structurally the same as the logistic model P 1 dp m e l 7 dt p m Thus K corresponds to l em and r corresponds to m 6 This model was motivated by an insect pest control problem populations over a wide area uctuated in asynchrony Hence to eradicate the pest you have to treat all areas simultaneous because if you ignore the areas not currently a problem they eventually recolonize all the other patches quotAs a conceptual and mathematical tool the Levins model was something new in population ecology and a necessary first step towards quantitative research in this areaquot Hanski and Gilpin l99l5 What are some reasonable time scales for this kind of model What factors affect the parameters 6 and m Body size Reproduction and survival rate FW662 Lecture 8 Spatially structured populations 5 Extension to this model Hanski 1991 de ned m as a function of distance isolation from other patches where D is average distance between patches and mg and a are parameters Likewise with eg and b parameters andA the average patch area Then the equilibrium proportion of patches occupied is e p1 oerbAaD m0 This extension still doesn39t take into account individual patch characteristics ie patch heterogeneity Hastings 1991 has published much more complex models However he has made no attempt to relate them to biology again probably because so little is known about extinction and recolonization rates Another extension of this model is to incorporate stochasticity to estimate metapopulation persistence times Hanski 1991 The lowest form of stochasticity is suggested by previous examples of stochastic population models Demographic stochasticity of patches or local populations which Hanski 1991 refers to as immigrationemigration stochasticity Still more stochasticity can be incorporated by variation in population dynamics across the patches which Hanski 1991 refers to as regional stochasticity Regional stochasticity results in the quotspreading of riskquot concept of den Boor 1968 quotSpecies with high dispersal rate but little regional stochasticity are expected to have long metapopulation persistence times while species with low dispersal rate but much regional stochasticity have short metapopulation persistence timesquot Hanski 1991 3334 Dispersal is most advantageous when their is little regional stochasticity and least advantageous when their is much regional stochasticity Gadgil 1971 in Hanski 1991 FW662 Lecture 8 Spatially structured populations 6 In summary 4 forms of stochasticity might be incorporated to achieve realistic estimates of metapopulation persistence 1 demographic stochasticity within patches 2 immigrationemigration stochasticity of patches 3 environmental stochasticity within patches and 4 regional stochasticity between patches In the following diagram demographic stochasticity would operate in each of the 4 patches immigrationemigration stochasticity would cause the immigration and emigration to vary randomly between the patches environmental stochasticity would operate in that each patch would vary through time but could all vary the same through time and regional stochasticity would cause the 4 patches to vary independently of each other Note that 2 additional sets of diagonal links between the patches have been left out of the diagram Stochasticity Demographic ImmigrationEmigration Environmental Temporal Regional Spatial Some causes of metapopulation extinction Hanski 19913435 1 e gt m ie there is no solution to Levins equation 2 alternative equilibria 3 immigrationemigration stochasticity ie all the patches go extinct at the same time 4 region stochasticity ie catastrophes that wipe out large portions of the range of the population Empirical observations to support spatial population models The main deviation from the theory is due to problems with local extinctions Populations tend not to wink on and off as portrayed by the metapopulation model Harrison 1991 suggests few empirical observations fit the Levins model well The more likely situations are 1 mainlandisland and sourcesink spatial populations in which persistence depends FW662 Lecture 8 Spatially structured populations 7 on the existence of one or more extinctionresistant populations As an example the checkered white butter y in Central Valley California as its source as riparian areas Some patches have 6 0 other patches have 6 gt 0 2 patchy populations in which dispersal between patches or subpopulations is so high that the system is effectively a single extinctionresistant population In this case m gtgt 6 3 nonequilibrium metapopulations in which local extinction occurs in the coarse of a species overall regional decline This suggests a modi ed view of metapopulation dynamics in which local extinction is more an incidental occurrence than a central feature A natural example of this phenomena are mammal populations in the sky island country of the southwest where mammals were isolated on mountain top habitat during postPleistocene warming A more frequent example is humancaused fragmentation of natural habitats Hanski 1994 describes a procedure to fit observed metapopulation data with maximum likelihood The model is fitted to presenceabsence data from a metapopulation at dynamic equilibrium between extinctions and colonizations However the con dence intervals on all 3 of the parameters estimated from the data include zero suggesting that in fact the model provides an uninformative fit to the observed data Spatial population theory and conservation biology Habitat fragmentation is making populations resemble metapopulation theory If individuals routinely move between habitat fragments then we have a metapopulation Theory suggests that isolated patches should have some movement of individuals even if the movement is in the form of transplants as part of management SLOSS is a controversy about the design of reserves SLOSS is Single Large or Several Small reserves One large reserve suffers a greater chance of a single catastrophe wiping out the population In contrast the probability of extinction of a population in a single small patch may be quite high and hence several small patches may not provide longterm persistence Spatial population theory and harvest theory McCullough 1996 Spatial controls can achieve high yields and avoid the hazards of overharvest that are possible with harvest quotas without detailed population data Harvesting of metapopulations holds little prospect because of negative effects on dispersal required for recolonization of patches following local extinction Summary The concept of a metapopulation is imaginative and does offer some insight into how to manage populations given that habitat is being fragmented The models suggest some interesting hypotheses However real data don t appear to support the simplest models Literature Cited FW662 Lecture 8 Spatially structured populations 8 den Boer P J 1968 Spreading of risk and stabilization of animal numbers Acta Biotheoretica 18 165 194 Gill D E 1978 The metapopulation ecology of the redspotted newt Notophthalmus viridescens Rafmesque Ecological Monograph 48 145166 Gill D E 1987 Effective population size and interdemic migration rates in a metapopulation of the redspotted newt Evolution 32839849 Gilpin M E 1987 Spatial structure and population vulnerability Pages 125139 in M E Soule ed Viable Populations for Conservation Cambridge University Press New York New York USA 189 pp Hanski I 1996 Metapopulation ecology Pages 1343 in Rhodes 0 E Jr R K Chesser and M H Smith eds Population dynamics in ecological space and time University Chicago Press Chicago Illinois USA Hanski I 1994 A practical model of metapopulation dynamics J Animal Ecology 63151 162 Hanski I and M Gilpin 1991 Metapopulation dynamics brief history and conceptual domain Biological J Linean Soc 42316 Harrison S 1991 Local extinction in a metapopulation context an empirical evaluation Biological J Linean Soc 427388 Hastings A 1990 Spatial heterogeneity and ecological models Ecology 71426428 Hastings A 1991 Structured models of metapopulation dynamics Biological J Linean Soc 425771 Levins R 1969 Some demographic and genetic A for biological control Bull Ent Soc Amer 15237240 39 II 39 of y Levins R 1970 Extinction Lect Math Life Sci 275107 Lidicker W Z 1975 The role of dispersal in the demography of small mammals Pages 103 128 in F B Golley K Petrusewicz and L Ryszkowski eds Small Mammals Their Productivity and Population Dynamics Cambridge University Press Cambridge United Kingdom Pulliam H R 1996 Sources and sinks empirical evidence and population consequences Pages 4569 in Rhodes 0 E Jr R K Chesser and M H Smith eds Population FW662 Lecture 8 Spatially structured populations dynamics in ecological space and time University Chicago Press Chicago Illinois USA Pulliam H R 1988 Sources sinks and population regulation American Naturalist 132652 Van Horne B 1983 Density as a misleading indicator of habitat quality Journal of Wildlife Management 47 893901 Weins J A 1996 Wildlife in patch environments metapopulations mosics and management Pages 5384 in D R McCullough ed Metapopulations and wildlife conservation Island Press Washington DC USA Wright S 1943 Isolation by distance Genetics 28114138 FW662 Lecture 14 Estimating Variance Components Lecture 14 Estimating Variance Components Reading Burnham K P D R Anderson G C White C Brownie and K H Pollock 1987 Design and Analysis Experiments for Fish Survival Experiments Based on CaptureRecapture Am Fish Monograph No 5 Pages 260278 We have discussed various sources of variance that impact the dynamics of a population demographic environmental and spatial process individual heterogeneity and genetic variances In addition the concept of sampling variance or the uncertainty of our estimates of population parameters has been frequently mentioned This chapter covers the statistical methodology to estimate the different variance components from data Consider the example situation of estimating survival rates each year for 10 years from a deer population Each year the survival rate is different from the overall mean because of snow depth cold weather etc Let the true but unknown overall mean be S Then the survival rate for each year can be considered to be S some deviation Environmental Variation 139 Mean Year 139 Yearz39 l S S 6 S 2 S S 62 S2 3 S S 63 S3 4 S S 6 S4 5 S S 6 S5 6 S S 65 S5 7 S S 67 S7 8 S S 68 S8 9 S S eg S9 10 S S em Sm Mean S S S The estimator ofS is S FW662 Lecture 14 Estimating Variance Components 2 g 11 10 with the variances of the Si 10 2091 7 2 A2 11 10 where the random variables el are selected from a distribution with mean 0 and variance 02 In reality we are never able to observe the annual rates because of sampling variation or demographic variation For example even if we observed all the members of a population we would still not be able to say the observed survival rate was S because of demographic variation Consider ipping 10 pennies We know that the true probability of a head is 05 but we will not always observe that value exactly The same process operates in a population as demographic variation Even though the true probability of survival is 05 we would not necessarily see exactly 12 of the population survive on any given year Hence what we actually observe are the quantities Environmental Variation Sampling Variation 1 Mean Year 1 Year 1 S Se1f1 S Se2f2 S Se3f3 S Se4f4 S Se5f5 S Se5f5 S Se7f7 S S S S gtmgtmgtbgtwgtNgtjngt n gt Se8f8 Se9fg OOONONUIAUJN ooo gt o 8 Semfm Dbl Mean S where the ei are as before but we also have additional variation from sampling or demographic FW662 Lecture 14 Estimating Variance Components 3 variation The standard approach to estimating the sampling variance separately from the environmental variance is to take replicate observations within each year in this example so that the within cell replicates can be used to estimate the sampling variance whereas the between cell variance is used to estimate the environmental variation Years are assumed to be a random effect and mixed model analysis of variance procedures are used eg Bennington and Thayne 1994 This approach assumes that each cell has the same sampling variance An example of the application of a random effects model is Koenig et al 1994 They considered year effects species effects and individual tree effects The assumption of constant variance within cells across a variety of treatment effects is often not true ie the sampling variance of a binomial distribution is a function of the parameter estimate Another common violation of this assumption is caused by the variable of interest being distributed lognormally so that the coefficient of variation is constant across cells so that the cell variance is a function of the cell mean Further the empirical estimation of the variance from replicate measurements may not be the most efficient procedure Again the binomial estimator of a survival rate is a good example of this Therefore the remainder of this chapter describes methods which can be viewed as extensions of the usual variance component analysis based on replicate measurements within cells We will be examining estimators for the situation where the within cell variance is estimated by some other estimator than from the moment estimator based on replicate observations Assume that we can estimate the sampling variance for each year given a value of SI for the year For example the sampling variation for a binomial is A 1 7 S7 VarSilSi 7 where n is the number of animals monitored to see if they survived Then can we estimate the variance term due to environmental variation given that we have estimates of the sampling variance for each year If we assume all the sampling variances are equal the estimate of the overall mean is still just the mean of the 10 estimates with the theoretical variance being FW662 Lecture 14 Estimating Variance Components 7 2 A MKS 0 E1IarSlS ie the total variance is the sum of the environmental variance plus the expected sampling variance This total variance can be estimated as 10 A A 2 Si 7 2 A A i1 var S 1010 7 1 We can estimate the expected sampling variance as the mean of the sampling variances 10 A 2 vans S E var A H 53 10 so that the estimate of the environmental variance obtained by solving for O2 10 A A 10 A 2 SI 7 S2 E varSilSi A2 i1 7 i1 10 7 l 10 However normally the sampling variances are not all equal so that we have to weight them to obtain an unbiased estimate of Oz The general theory says to use a weight w l wi A 02 varSI lSI I 1 Z W 1 10 2w 1 so that the estimator of the weighted mean is 0 i with theoretical variance see Box 1 for a derivation of this result 1 var 10 FW662 Lecture 14 Estimating Variance Components and empirical variance estimator 10 Ewi i 7 2 11 10 Mg 10 7 1 i1 When the w are the true but unknown weights we have 10 Ewi i 7 2 11 10 21 10 7 1 i1 giving the following 10 A 7 Z wiSi 7 2 11 10 7 1 Hence all we have to do is manipulate this equation with a value of O2 that is imbedded in the witerm to obtain an estimator of 02 To obtain a con dence on the estimator of 02 we can substitute the appropriate chisquare values in the above relationship To nd the upper con dence interval value 6U solve the equation 10 Ewi i 532 11 7 X210 711xL 10 7 1 10 7 1 and for the lower con dence interval value 6 solve the equation 10 A A EwiSi 7 S2 2 H 7 X 10 7 LocU 10 7 1 7 10 7 1 As an example consider the following fawn survival data from Little Hills FW662 Lecture 14 Estimating Variance Components Estimated Estimated Year Collared Lived Survival Variance 81 46 15 03260870 00047773 82 114 38 03333333 00019493 83 118 5 00423729 00003439 84 106 19 01792453 00013879 85 155 59 03806452 00015210 86 161 61 03788820 00014617 87 116 15 01293103 00009706 The survival rates are the number of collared animals that lived divided by the total number of collared animals For 1981 S1981 1546 0326087 The sampling variance associated with this estimate computed as A s 1 7 s Var S 1981 1981 1981 46 which equals 00047773 The spreadsheet VARCOMP WBZ computes 2 as 00170632 6 01306262 with a 95 con dence interval of 00064669 00869938 for 02 and 00804167 02949472 for 0 Sensitivity of the Leslie matrix elements is inversely related to the process variance of the life history traits P ster 1998 That is Sensitivity Variance of matrix entry aij FW662 Lecture 14 Estimating Variance Components 7 because the process variance of A is a function of the process variance of the parameter 6A 2 vart a varaij 1 assuming no covariances between the ai elements In order to persist a population must have a limited amount of variation Thus natural selection will select against high process variance in a parameter that A is very sensitive too Literature Cited Bennington C C and W V Thayne 1994 Use and misuse of mixed model analysis of variance in ecological studies Ecology 75717722 Burnham K P D R Anderson G C White C Brownie and K H Pollock 1987 Design and Analysis Experiments for Fish Survival Experiments Based on CaptureRecapture Am Fish Monograph No 5 Pages 260278 Burnham K P and G C White 2002 Evaluation of some random effects methodology applicable to bird ringing data Journal of Applied Statistics 2914245264 Koenig W D R L Mumme W J Carmen and M T Stanback 1994 Acorn production by oaks in central California variation within and among years Ecology 7599109 Pf1ster C A 1998 Patterns of variance in stagestructured populations evolutionary predictions and ecological implications Proceedings of National Academy of Science 95213218 FW662 Lecture 14 Estimating Variance Components Box 1 Derivation of the result that Starting with the result that 4 02 var ilSi var i Wi the derivation is as follows var va H 10 wi i1 10 Var wiSI 10 2 i1 wi 1 10 A va wS 10 2H i I 39 wi i1 1 10 Z wizvarS 10 391 wi 1 10 w 10 211 39 Z W i1 7 1 10 Z W FW662 Lecture 14 Estimating Variance Components FW662 Lecture 15 PVA 1 Lecture 15 Minimum Viable Population Models Estimating Population Persistence Probabilities Review Reading Beissinger S R and M I Westphal 1998 On the use of demographic models of population Viability in endangered species management Journal of Wildlife Management 62821841 Optional Boyce M S 1992 Population Viability analysis Annual Review of Ecology and Systematics 23481506 A stande de nition of a population is a group of individuals of the same species occupying a de ned area at the same time Hunter 1996 Two procedures are commonly used for evaluating the Viability of a r r 39 quot or the r 39 39 quotquot that the r r 39 quot will survive for some speci ed time Population Viability analysis PVA is the methodology of estimating the probability that a population of a speci ed size will persist for a speci ed length of time The minimum Viable population MVP is the smallest population size that will persist some speci ed length of time with a speci ed probability In the rst case the probability of extinction is estimated whereas in the second the number of animals is estimated that is needed in the population to meet a speci ed r 39 39 quotquot ofr 39 For a 39 that is expected to go extinct the time to extinction is the expected time the population will persist Both PVA and MVP require a time horizon ie a speci ed but arbitrary time to which the probability of extinction pertains rr The topic of PVA has become very popular with 2 recent books Beissinger and McCullough 2002 Morris and Doak 2002 providing extensive coverage of the topic De nitions and criteria for Viability persistence and extinction are arbitrary e g a 95 probability of a population persisting for at least 100 years Boyce 1992 Mace and Lande 1991 discuss criteria for extinction Ginzburg et al 1982 suggest the phrase quasiextinction risk as the probability of a population dropping below some critical threshold a concept also promoted by Morris and Doak 2002 Ludwig 1996a and Dennis et al 1991 Schneider and Yodzis 1994 use the term quasiextinction to mean the population dropped to only 20 females remaining The usual approach for estimating persistence is to develop a probability distribution for the number of years before the model quot goes extinctquot or below a speci ed threshold The percentage of the area under this distribution where the population persists beyond a speci ed time period is taken as an estimate of persistence To obtain MVP probabilities of extinction are needed for various initial population sizes The expected time to extinction is a misleading indicator of population Viability Ludwig 1996b because for small populations the probability of extinction in the immediate future is high even though the expected time until extinction may be quite FW662 Lecture 15 PVA 2 large The skewness of the distribution of time until extinction thus makes the probability of extinction for a speci ed time interval a more realistic measure of population viability Simple stochastic models have yielded qualitative insights into population viability questions Dennis et al 1991 But because population growth is generally considered to be nonlinear with nonlinear dynamics making most stochastic models intractable for analysis Ludwig 1996b and because catastrophes and their distribution pose even more difficult statistical problems Ludwig 1996b analytical methods are generally inadequate to compute these probabilities Hence computer simulation is commonly used to produce numerical estimates for persistence or MVP Analytical models lead to greater incites given the simplifying assumptions used to develop the model However the simplicity of analytical models precludes their use in real analyses because of the omission of important processes governing population change such as age structure and periodic breeding Lack of data suggests the use of simple models but lack of data really means lack of information Lack of information suggests that no valid estimates of population persistence are possible since there is no reason to believe that unstudied populations are inherently simpler and thus justify simple analytical models than wellstudied populations where the inadequacy of simple analytical models is obvious The focus of this paper is on computer simulation models to estimate population viability via numerical techniques where the population model includes the essential features of population change relevant to the species of interest The most thorough recent reviews of the PVA literature are provided by Beissinger and Westphal 1998 and Boyce 1992 Shaffer 1981 1987 Soule 1987 Nunney and Campbell 1993 and Remmert 1994 provide an historical perspective of how the field developed Q quot I 39 r 39 quot 39 39 39 39 know a 39J 39 39 amount about what allows populations to persist Some generalities about population persistence Ruggiero et al 1994 are 1 connected habitats are better than disjointed habitats 2 suitable habitats in close proximity to one another are better than widely separated habitats 3 late stages of forest development are often better than younger stages 4 larger habitat areas are better than smaller areas 5 populations with higher reproductive rates are more secure than those with lower reproductive rates and 6 environmental conditions that reduce carrying capacity or increase variance in the growth rates of populations decrease persistence probabilities This list should be taken as a general set of principles but you should recognize that exceptions will occur often In the following section I will discuss these generalities in more detail and in particular suggest contradictions that occur FW662 Lecture 15 PVA 3 Typically recovery plans for an endangered species try to 1 create multiple populations of the species so that a single catastrophe will not wipe out the entire species and 2 increase the size of each population so that genetic demographic and normal environmental uncertainties are less threatening Meffe and Carroll 1994 191192 However Hess 1993 argues that connected populations can have lower viability over a narrow range in the presence of a fatal disease transmitted by contact He demonstrates the possibilities with a model but doesn39t have data to support his case However the point he makes seems biologically sound and the issue can only be resolved by optimizing persistence between these two opposing forces Spatial variation ie variation in habitat quality across the landscape affects population persistence Typically extinction and metapopulation theories emphasize that stochastic uctuations in local populations cause extinction and that local extinctions generate empty habitat patches that are then available for recolonization Metapopulation persistence depends on the balance of extinction and colonization in a static environment Hanski 1996 Hanski et al 1996 For many rare and declining species Thomas 1994 argues 1 that extinction is usually the deterministic consequence of the local environment becoming unsuitable through habitat loss or modification introduction of a predator etc 2 that the local environment usually remains unsuitable following local extinction so extinctions only rarely generate empty patches of suitable habitat and 3 that colonization usually follows improvement of the local environment for a particular species or longdistance transfer by humans Thus persistence depends predominantly on whether organisms are able to track the shifting spatial mosaic of suitable environmental conditions or on maintenance of good conditions locally Foley 1994 uses a model to agree with 5 above that populations with higher reproductive rates are more persistent However mammals with larger body size can persist at lower densities Silva and Downing 1994 and typically have lower annual and per capita reproductive rates The predicted minimal density decreases as the 068 power of body mass likely because of less variance in reproduction relative to life span The last item on the list above suggests that increased variation in time leads to lower persistence Shaffer 1987 Lande 1988 1993 One reason that increased temporal variation causes lowered persistence is that catastrophes such as hurricanes fires or floods are more likely to occur in systems with high temporal variation Populations in the wet tropics can apparently sustain themselves at densities much lower than those in temperate climates likely because of less environmental variation Basically the distinction between a catastrophe and a large temporal variance component is arbitrary and on a continuum Caughley 1994 Further even predictable effects can have an impact Beissinger 1995 models the effects of periodic environmental uctuations on population viability of the snail kite Rostrhamus sociabilis Few empirical data are available to support the generalities above but exceptions exist Berger 1990 addressed the issue of MVP by asking how long differentsized populations persist He presents demographic and weather data spanning up to 70 years for 122 bighom sheep Ovis canadensis populations in southwestem North America His analyses reveal that l 100 FW662 Lecture 15 PVA 4 percent of the populations with fewer than 50 individuals went extinct within 50 years 2 populations with greater than 100 individuals persisted for up to 70 years and 3 the rapid loss of populations was not likely to be caused by food shortages severe weather predation or interspecific competition Thus 50 individuals even in the short term of 50 years are not a minimum viable population size for bighom sheep However Krausman et al 1993 questioned this result because they know of populations of 50 or less in Arizona that have persisted Pimm et al 1988 and Diamond and Pimm 1993 examined the risks of extinction of breeding land birds on 16 British islands in terms of population size and species attributes Tracy and George 1992 extended the analysis to include attributes of the environment as well as species characteristics as potential determinants of the risk of extinction Tracy and George 1992 conclude that the ability of current models to predict the risk of extinction of particular species on particular island is very limited They suggest models should include more speci c information about the species and environment to develop useful predictions of extinction probabilities Haila and Hanski 1993 criticized the data of Pimm et al 1988 as not directly relating to extinctions because the small groups of birds breeding in any given year on single islands were not populations in a meaningful sense Although this criticism may be valid most of the populations that conservation biologists will study will be questionable populations Thus results of the analysis by Tracy and George 1992 do contribute useful information Specifically small populations of smallbodied birds on oceanic islands more isolated are more likely to go extinct than are large populations of largebodied birds on less isolated channel islands However interaction of body size with type of island channel vs oceanic indicated that body size in uences time to extinction differently depending on the type of island The results of Tracy and George 1992 1993 support the general statements presented above As with all ecological generalities exceptions quickly appear Typically extinction and metapopulation theories emphasize that stochastic uctuations in local populations cause extinction and that local extinctions generate empty habitat patches that are then available forr 39 39 quot r r 39 im 1 39 depends on the balance of extinction and colonization in a static environment For many rare and declining species Thomas 1994 argues 1 that extinction is usually the deterministic consequence of the local environment becoming unsuitable through habitat loss or modification introduction of a predator etc 2 that the local environment usually remains unsuitable following local extinction so extinctions only rarely generate empty patches of suitable habitat and 3 that colonization usually follows improvement of the local environment for a particular species or longdistance transfer by humans Thus persistence depends predominantly on whether organisms are able to track the shifting spatial mosaic of suitable environmental conditions or on maintenance of good conditions locally Many factors affect the persistence of a population What components are needed to provide estimates of the probability that a population will go extinct and what are the tradeoffs if not all these components are available FW662 Lecture 15 PVA 5 l A basic population model is needed A recognized mechanism of population regulation density dependence should be incorporated because no population can grow inde nitely quotOf course exponential growth models are strictly unrealistic on time scales necessary to explore extinction probabilitiesquot Boyce 1992489 The population cannot be allowed to grow inde nitely or persistence will be over estimated Further as discussed below the shape of the relationship between density and survival and reproduction and can affect persistence and density dependence cannot be neglected for moderate or large populations Ludwig 1996b Density dependence can provide a stabilizing in uence that increases persistence in small populations Demographic variation must be incorporated in this basic model Otherwise estimates of persistence will be too high because the effect of demographic variation for small populations is not included in the model Temporal variation must be included for the parameters of the model including some probability of a natural catastrophe Examples of catastrophes are fires e g Yellowstone National Park USA during 1988 hurricanes typhoons earth quakes extreme drought or rainfall resulting in ooding etc Catastrophes must be rare or else the variation would be considered as part of the normal temporal variation However the covariance of the parameters is also important Good years for survival are likely also good years for reproduction Vice versa bad years for reproduction may also lead to increased mortality The impact of this correlation of reproduction and survival can drastically affect results For example the model of Stacey and Taper 1992 of acorn woodpecker population dynamics performs very differently depending on whether adult survival juvenile survival and reproduction are bootstrapped as a triplet or as individual rates across the 10 year period By allowing the correlation of the survival rates and reproduction persistence is improved mainly because the effects of one year in the data with both low juvenile survival and low reproduction is somewhat ameliorated by always combining these 2 rates Spatial variation in the parameters of the model must be incorporated if the population is spatially segregated If spatial attributes are to be modeled then immigration and emigration parameters must be estimated as well as dispersal distances The difficulty of estimating spatial variation is that the covariance of the parameters must be estimated as a function of distance ie what is the covariance of adult survival of 2 subpopulations as a function of distance Individual heterogeneity must be included in the model Individual heterogeneity requires that the basic model be extended to an individualbased model DeAngelis and Gross 1992 As the variance of individual parameters increases in the basic model the persistence time increases Conner and White 1999 White FW662 Lecture 15 PVA 6 2000 Thus instead of just knowing estimates of the parameters of our basic model we also need to know the statistical distributions of these parameters across individuals This source of variation is not mentioned in discussions of population viability analysis eg Boyce 1992 Remmert 1994 Hunter 1996 Meffe and Carroll 1994 or Shaffer 1981 1987 However recent articles Kendall and Fox 2002 Fox and Kendall 2002 recognize individual heterogeneity and in particular discuss how individual heterogeneity reduce the impacts of demographic stochasticity in PVAs An important consequence is that almost all PVA overestimate the importance of demographic stochasticity and therefore the risk of extinction Fox and Kendall 2002 Many studies have demonstrated individual heterogeneity of individual survival and reproductions e g CluttonBrock 1982 demonstrated lifetime reproductive success of female red deer Cervus elaphus varied from 0 to 13 calves reared per female Differences in the frequency of calf mortality between mothers accounted for a larger proportion of variance in success than differences in fecundity Suppose the population has a large variance of adult survival ie some adults have very high survival whereas other have much lower survival Assume that adult survival rates are an individual characteristic ie an individual s survival rate might uctuate with temporal variation but individuals with high survival will always have higher survival than individuals with low survival rates Compare this situation to the typical model where all animals have the same survival rate We find that persistence is greatest in populations with high variation of basic population parameters because some individuals have much greater survival potential than average and thus are not removed from the population at the average rate A third r quot quotquot is that 39 quot 39 39 39 39 quot exists in the population but the relative survival rates do not endure across time That is an individual with high survival in year 1 may have the lowest survival in year 2 Because of this random ucuation across years each individual s expected survival probability across several years would be the same but with more variation than if each had the same survival rate The effect of each of these 3 assumptions is shown in the following graph of a simple death process FW662 Lecture 15 PVA 7 Death Process with Ind Heterogeneity 100 Population Time As the variance of individual parameters increases in the lifelong model the persistence time increases The above graph was generated assuming average annual mortality was 01 For the no heterogeneity model each animal had probability 09 of surviving 1 additional year In the annual variation model an animal s annual mortality rate was selected from a beta distribution with ac l and 5 9 giving a mean of a 01 and variance 06 3 l 06B20 l51 0 mode only for oz 2 1 For the life 000818 with the mode 1 CC 5 7 2 long model an animal s lifelong mortality rate was selected from a beta distribution with ac l and 5 9 giving the same mean and variance as for the annual rate model If the beta distribution parameters are changed to increase the variance in the lifelong model eg oc 05 and 5 45 then even greater persistence is achieved In contrast changing the beta distribution parameters of the annual rate model to these same values only increases the variance of the estimated persistence time not the expected value 6 For shortterm projects the above sources of variation may be adequate However if time periods of more than a few generations are projected then genetic variation should be considered I would expect the population to change as selection takes place Even if no selection is operating then genetic drift is FW662 Lecture 15 PVA 8 expected for small population sizes However the importance of genetic effects is still an issue in question e g Joopouborg and Van Groenendael 1996 Lande 1988 1995 has suggested either demographic variation andor genetic effects can be lethal to a small population 7 For longterm persistence we must be willing to make the assumption that the system will not change ie the levels of stochasticity will not change through time the species will not evolve through selection and the supporting capacity of the environment the species habitat remains static That is natural processes such as longterm succession and climatic change do not affect persistence and that humans cease and desist given that humans have been responsible for most recent extinctions To believe the results we have to assume that the model and all its parameters stays the same across inordinately long time periods After examining this list I am sure you agree with Boyce 1992482 quotCollecting sufficient data to derive reliable estimates for all the parameters necessary to determine MVP is simply not practical in most casesquot Of course limitations of the data seldom slow down modelers of population dynamics Further managers are forced to make decisions so modelers attempt to make reasonable quotguessesquot ESTIMATION OF VARIANCE COMPONENTS The implication of the list of requirements in the previous section is that population parameters or their distributions are known without error ie exact parameter values are observed not estimated In reality we may be fortunate and have a series of survival or reproduction estimates across time that provides information about the temporal variation of the process However the variance of this series is not the proper estimate of the temporal variation of the process This is because each of our estimates includes sampling variation ie we only have an estimate of the true parameter not its exact value To properly estimate the temporal variation of the series the sampling variance of the estimates must be removed The previous lecture demonstrated a technique to remove sampling variance from a series of estimates ie a method to estimate the process variance A second approach is to incorporate additional information from covariates Individual heterogeneity occurs in both reproduction and survival Estimation of individual variation in reproduction is an easier problem than estimation of individual variation in survival because some animals reproduce more than once whereas they only die once Bartmann et al 1992 demonstrated that overwinter survival of mule deer fawns is related to their weight at the start of the winter Thus one approach to modeling individual heterogeneity is to find a correlate of survival that can be measured and develop statistical models of the distribution of this correlate Then the distribution of the correlate can be sampled to obtain an estimate of survival for the individual Lomnicki 1988 also suggests weight as an easily measured variable that relates to an animal s fitness FW662 Lecture 15 PVA 9 To demonstrate this methodology I will use a simpli cation of the logistic regression model of Bartmann et al 1992 S log 50 BIWeight where survival S is predicted as a function of weight Weight of fawns at the start of winter was approximately normally distributed with mean 32 kg and standard deviation 42 To simulate individual heterogeneity in overwinter fawn survival values can be drawn from this normal distribution to generate survival estimates This model can be expanded to incorporate temporal variation year effects sex effects and area effects as described for mule deer fawns by Bartmann et al 1992 An example of modeling temporal variation in greater amingos Phoenicopterus ruber roseus as a function of winter severity is provided by Cezilly et al 1996 The approach suggested here of modeling winter severity as a random variable and estimating survival as a function of this random variable is an alternative to the variance estimation procedures of the previous section Both provide a mechanism for injecting variation into a population viability model The main advantage of using weather data to drive the temporal variation of the model is that considerably more weather data is available than is biological data on survival or reproductive rates The major drawback of the indirect estimation approach proposed here is that sampling variation of the functional relationship is ignored in the simulation procedure That is the logistic regression model includes sampling variation because its parameters are estimated from observed data The parameter estimates of the logistic regression model include some unknown estimation error Their direct use results in potentially biased estimates of persistence depending on how much sampling error is present Thus a good model relating the covariate to the biological process is needed A third method is demonstrated by Stacy and Taper 1992 when they used a bootstrap procedure to incorporate temporal variation into a model of acorn woodpecker M elanerpes formicz39vorus population viability They used estimates of adult and juvenile survival and reproductive rates resulting from a 10year field study to estimate population persistence To incorporate the temporal variation from the 10 years of estimates they randomly selected with replacement 1 estimate from the observed values to provide an estimate in the model for a year This procedure is known in the statistical literature as a bootstrap sampling procedure The technique is appealing because of its simplicity However for estimating population viability a considerable problem is inherent in the procedure That is the estimates used for bootstrapping contain sampling variation and demographic variation as well as the environmental variation which the modeler is attempting to incorporate To illustrate how demographic variation is included in the estimates consider an example population of 10 FW662 Lecture 15 PVA 10 animals with a constant survival rate of 055 Thus the actual temporal variation is zero yet a sequence of estimates of survival from this population would suggest considerable variation That is the estimates of survival would have a variance of 0551 05510 002475 if all 10 animals had a survival probability of 055 Further the only observed values of survival would be 0 01 10 However ifthe size ofthe population is increased to 100 you nd that the variance of the sequence of estimates is now 0002475 a considerable decrease from above Thus randomly sampling the estimates from a population of size 10 results in considerably more variation than from a population of 100 As a result the demographic variation from the sampled population will be incorporated into the persistence model if the bootstrap approach is used A similar example can be used to demonstrate that sampling variation is also inherent in bootstrapping from a sample of observed estimates Suppose a sample of 10 radiocollared animals is used to estimate survival for a population of 100000 animals ie the nite sample correction term can be ignored The sampling variation of the estimates would be S1 S 10 where S is the true survival rate for the population assuming all animals had the same survival rate Now if a sample of 100 radiocollared animals is taken the sampling variation reduces to S1 S 100 Thus randomly sampling estimates with a bootstrap procedure incorporates the sampling variation of the estimates into the persistence model As a result of the increased variation persistence values will be underestimated Therefore I suggest using sparingly the bootstrap approach demonstrated by Stacey and Taper 1992 Persistence estimates developed with this procedure will generally be too low ie you will conclude the population is more likely to go extinct than it really will However methodologies such as shrinkage estimation of variances K P Bumham Pers Commun may prove useful in removing sampling variance from the estimates and make the bootstrap procedure more applicable to estimating population persistence INCORPORATION OF PARAMETER UNCERTAINTY INTO PERSISTENCE ESTIMATES Unbiased estimates of process variances such as temporal and spatial variation can be achieved In this section I will examine how to incorporate uncertainty of the parameter estimates into the estimates of persistence and in the process provide an unbiased estimate of persistence given the population model Any model developed to estimate population persistence will have several to many parameters that must be estimated from available data Each of these estimates will have an associated estimate of its precision in the from of a variance assuming that statistically rigorous methods were used to estimate the parameter from data In addition because some of the parameters may have been estimated from the same sets of data some parameters in the model may have a non zero covariance Thus the vector of parameter estimates Q used in the model to estimate persistence has the variancecovariance matrix Var to measure uncertainty FW662 Lecture 15 PVA 11 Typically statisticians use the delta method eg Seber 198279 to estimate the variance of a function of parameters from a set of parameter estimates and their variancecovariance matrix In the context of persistence the variance of the estimate of persistence g3 would be estimated as a 0 T A a 0 V rQ LEE V r where g3 That is the function f represents the model used to estimate persistence However for realistically complex persistence models the analytical calculation of partial derivatives needed in this formula is likely not feasible The lack of explicit analytical partial derivatives suggests that numerical methods be used The most feasible albeit numerically intensive appears to be the parametric bootstrap approach Effron and Tibshirani 1993 Urban Hjorth 1994 With a parametric bootstrap a realization of the parameter estimates is generated based on their point estimates and sampling variance covariance matrix using Monte Carlo methods Likely a multivariate normal distribution will be used as the parametric distribution describing the set of parameter estimates although other distributions or combinations of distributions may be more realistic biologically Using this set of simulated values in the persistence model persistence is estimated This step will require a large number of simulations to properly estimate persistence with little uncertainty typically 10000 simulations are conducted Then a new set of parameter values are generated and persistence again estimated This process is repeated for many sets of parameter estimates at least 100 but more likely 1000 to obtain a set of estimates of persistence The variation of the resulting estimates of persistence is then a measure of uncertainty attributable to the variation of the parameter estimates as measured by their variancecovariance matrix The process is diagramed as PARAMETRIC BOOTSTRAP LOOP 1000 iterations Select realization of parameter estimates MONTE CARLO LOOP 10000 iterations Tabulate percentage of model runs resulting in persistence END MONTE CARLO LOOP END PARAMETRIC BOOTSTRAP LOOP FW662 Lecture 15 PVA 12 However even more critical to our viability analysis is the fact that the mean of this set of 1000 estimates of persistence is likely less than the estimate we obtained using our original point estimates of model parameters More formally the expected value of estimated persistence E is less than the value of persistence predicted by our model using the point estimates of its parameters ie E ltfE an example of Jensen s inequality This difference is due to large probabilities of early extinction for certain parameter sets that are likely given their sampling variation Ludwig 1996a Thus to estimate persistence the mean of the bootstrap estimates of persistence should be used and not the estimate of persistence obtained by plugging our parameter estimates directly into our population model Confidence intervals on persistence could be constructed using the usual iZSE procedure based on the set of 1000 estimates This confidence interval represents the variation attributable to the uncertainty of the parameter estimates used in the model Uncertainty about the model is not included in this confidence interval because the model is assumed to be known However a better confidence interval will probably be achieved by sorting the 1000 values into ascending order and using the 253911 and 975 11 values as a 95 confidence interval This procedure accounts for the likely asymmetric distribution of the estimates of persistence DISCUSSION The real problem with PVA is not the model but obtaining the data to drive the models Ruggiero et al 1994 Ludwig 1999 Much of the published work on PVA ignores this essential Thomas 1990 For example Mangel and Tier 1994 simplify the process to the point that they miss major issues concerning data reliability and quality of the product estimates of persistence Their 4 facts are l quotA population can grow on average exponentially and without bound and still not persistquot This is because of catastrophes that will bring even a thriving population to zero 2 quotThere is a simple and direct method for the computation of persistence times that virtually all biologists can usequot They suggest a simple model with one age class and a population ceiling that the population cannot exceed but the ceiling does not cause density dependence effects of growth parameters As a result their approach to estimating persistence is likely to underestimate persistence if the ceiling is set too low because the population can never grow away from the absorbing state of extinction 3 quotThe shoulder of the MacArthurWilson model occurs with other models as well but disappears when catastrophes are includedquot They suggest a slow steady rise in persistence times as the population ceiling is increased FW662 Lecture 15 PVA 13 4 quotExtinction times are approximately exponentially distributed and this means that extinctions are likelyquot Thus they conclude the most likely value of a population is zero ie the mode of an exponential distribution I believe this result is because of the simplistic assumptions they have used to obtain it Realistic models that incorporate the sources of variation described above will not result in such simplistic results Another misguided example is Tomiuk and Loeschcke 1994 Their mathematics cover up the real problem of obtaining realistic estimates of the parameter values to use in the models Their model emphasizes demographic variation and ignores the bigger issues of temporal variation and individual heterogeneity A common problem with PVA is that the sampling variation of the parameter estimates is ignored Examples are Stacey and Taper 1993 and Dennis et al 1991 In both cases estimates of persistence are too pessimistic because the sampling variation of the population parameters is included in the population model as if it were temporal variation quotMost PVAs have ignored fundamentals of ecology such as habitat focusing instead on genetics or stochastic demographyquot Boyce 1992491 For small populations lt50 of endangered species such a strategy may be justified particularly for short term predictions But incorporating only demographic variation results in over estimates of persistence because temporal variation has been ignored On the other hand the remaining survivors of an endangered species may be the individuals with strong survival and reproductive rates and so the lack of individual heterogeneity may under estimate persistence The above studies should not lead the reader to believe that useful attempts to estimate persistence do not exist Schneider and Yodzis 1994 developed a model of marten M artes americana population dynamics that incorporated the behavior and physiology of individual martens spatial dynamics and demographic and environmental stochasticity Undoubtedly some readers would quibble with some of the assumptions and data used to build the model but I would contend that a realistic model with some of the inputs guessed is a much more reasonable approach than a simplistic model that ignores important processes affecting persistence Further such realistic models identify data needs that can be addressed with time even though the actual estimate of persistence is of questionable value The alternative of using simplistic and naive models assures invalid estimates and little progress in improving the situation with a rapid loss of credibility by the field of conservation biology Murphy et al 1990 have proposed two different types of PVA For organisms with low population densities that are restricted to small geographic ranges typical vertebrate endangered species genetic and demographic factors should be stressed For smaller organisms such as most endangered invertebrates environmental uncertainty and catastrophic factors should be stressed because these organisms are generally restricted to a few small habitat patches but are capable of reaching large population sizes within these patches Nunney and Campbell 1993 FW662 Lecture 15 PVA 14 note that demographic models and genetic models both have resulted in similar estimates of minimum viable population size but that the ideal spatial arrangement of reserves remains an issue Lande 1995 suggests that genetic mutations may affect tness and thus ignoring genetic effects results in underestimates of viability Mutation can affect the persistence of small populations by causing inbreeding depression by maintaining potentially adaptive genetic variation in quantitative characters and through the erosion of fitness by accumulation of mildly detrimental mutations Populations of 5000 or more are required to maintain evolutionary viability Theoretical results suggest that the risk of extinction due to the xation of mildly detrimental mutations may be comparable in importance to environmental stochasticity and could substantially decrease the longterm viability of populations with effective sizes as large as a few thousand Lande 1995 If these results are correct determining minimum viable population numbers for most endangered species is an exercise in futility because almost all of these populations are already below 5000 Conservation biologists would like to have rules of thumb to evaluate persistence Boyce 1992 for example the magical FranklinSoule number of 500 Franklin 1980 Soule 1980 that is the effective population size Ne to maintain genetic variability in quantitative characters Unfortunately these rules lack the realism to be useful The FranklinSoule number was derived from simple genetic models and hence lacks the essential features of a PVA model discussed here Attempts with simplistic models such as Mangel and Tier 1994 and Tomiuk and Loeschcke 1994 also do not provide defensible results because of the lack of attention to the biology of the species and the stochastic environment in which the population exists Until conservation biologists do good experimental studies to evaluate population persistence empirically I question the usefulness of rules of thumb and simplistic models suggested various places in the literature In the meantime until rigorous experimental work can be conducted conservation biologists should borrow information from game species where longterm studies have been done that will provide estimates of temporal and spatial variation and individual heterogeneity Rules of thumb that predict temporal variation in survival as a function of weather or individual variation in survival as a function of body characteristics provide alternative sources of data For at least some game species data exist to develop such rules Further these kinds of data will probably never be available for many endangered species the opportunity to collect such data was lost with the decline of the population to current threatened levels Thus I suggest the use of surrogate species to help meet the data needs of realistic models of persistence Taxonomically related species may provide information although species in the same ecological guild may also provide information on temporal and spatial variation CONCLUSION FW662 Lecture 15 PVA 15 In summary most estimates of population viability are nearly useless because one or more of the following mistakes or omissions are made in developing a model to estimate persistence l The model ignores spatial variation which will increase population viability As suggested by Stacey and Taper 1992 immigration can occasionally rescue a population from extinction The model uses estimates of temporal variation that are at best poor guesses This statement assumes that the modeler understood the difference between process variation and sampling variation Often sampling variation is assumed to substitute for process variation and as a result the estimates of persistence are too pessimistic Sampling variation has nothing to do with population persistence Estimates of population parameters must not be treated as if they are the true parameter value The model uses demographic variation as a substitute for temporal variation in the process and ignores true temporal variation The model ignores lifelong individual heterogeneity that increases population viability and assumes that all individuals endure the same identical survival and reproduction parameters Such a naive assumption results in population viability being underestimated The model assumes that current conditions are not changing ie the stochastic processes included in the model are assumed constant for the indefinite future Loss of habitat and other environmental changes that affect these stochastic processes are ignored Thus as discussed by Caswell 1989 the model is likely not useful in forecasting ie predicting what will happen but is useful in projecting ie predicting what would happen if conditions do not change Before you use the estimates of persistence from any population viability analysis compare your approach to obtain the estimate against the necessary components discussed here If you discover omissions and errors in the approach used to obtain the estimate recognize the worth or lack thereof of the estimate of persistence Although the estimates of persistence obtained from a PVA may have little value the process of formulating a model and identifying missing information ie parameters that are poorly estimated may still have value in developing measures to conserve the species in question Beissinger and Westphal 1998 Literature Cited Bartmann R M G C White and L H Carpenter 1992 Compensatory mortality in a Colorado mule deer population Wildlife Monograph 121 139 FW662 Lecture 15 PVA 16 Beissinger S R 1995 Modeling extinction in periodic environments Everglades water levels and Snail Kite population viability Ecological Applications 5618631 Beissinger S R and M I Westphal 1998 On the use of demographic models of populatioon viability in endangered species management Journal of Wildlife Management 62821 Beissinger S R and D R McCullough editors 2002 Population viability analysis University of Chicago Press Chicago Illinois USA 577 pp Berger J 1990 Persistence of differentsized populations an empirical assessment of rapid extinctions in bighorn sheep Conservation Biology 49198 Boyce M S 1992 Population viability analysis Annual Review of Ecology and Systematics 23 48 1 506 Caswell H 1989 Matrix population models Sinauer Associates Sunderland Massachusetts USA 328 pp Caughley G 1994 Directions in conservation biology Journal of Animal Ecology 63215244 Cezilly F A Viallefont V Boy and A R Johnson 1996 Annual variation in survival and breeding probability in greater amingos Ecology 77 11431 150 Clark T W P C Paquet and A P Curlee 1966 Special section large carnivore conservation in the Rocky Mountains of the United States and Canada Conservation Biology 936 936 CluttonBrock T H F E Guinness and S D Albon 1982 Red deer behavior and ecology of two sexes University Chicago Press Chicago Illinois 378 pp Conner M M and G C White 1999 Effects of individual heterogeneity in estimating the persistence of small populations Natural Resource Modeling 12109127 DeAngelis D L and L J Gross eds 1992 Individualbased models and approaches in ecology populations communities and ecosystems Chapman amp Hall New York New York USA 525 pp Dennis B P L Munholland and J M Scott 1991 Estimation of growth and extinction parameters for endangered species Ecological Monographs 6 1 15143 Diamond J and S Pimm 1993 Survival times of bird populations a reply American Naturalist 14210301035 FW662 Lecture 15 PVA 17 Efron B and R J Tibshirani 1993 An introduction to the bootstrap Chapman amp Hall New York New York USA 436 pp Foley P 1994 Predicting extinction times from environmental stochasticity and carrying capacity Conservation Biology 8 124136 Fox G A and B E Kendall 2002 Demographic stochasticity and the variance reduction effect Ecology 83 19281934 Franklin I R 1980 Evolutionary changes in small populations Pages 135149 In M E Soule and B A Wilcox eds Conservation biology an evolutionaryecological perspective Sinauer Associates Sunderland Massachusetts USA Ginzburg L R L B Slobodkin K Johnson A G Bindman 1982 Quasiextinction probabilities as a measure of impact on population growth Risk Analysis 2171181 Haila Y and I K Hanski 1993 Birds breeding on small British islands and extinction risks American Naturalist 142 10251029 Hanski I 1996 Metapopulation ecology Pages 1343 in Population Dynamics in Ecological Space and Time 0 E Rhodes Jr R K Chesser and M H Smith eds University Chicago Press Chicago Illinois USA Hanski I A Moilanen and M Gyllenberg 1996 Minimum viable metapopulation size American Naturalist 147527541 Hess G R 1993 Conservation corridors and contagious disease a cautionary note Conservation Biology 8256262 Hunter M L Jr 1996 Fundamentals of conservation biology Blackwell Science Cambridge Massachusetts USA 482 pp Joopouborg N and J M Van Groenendael 1996 Demography genetics or statistics comments on a paper by Heschel and Paige Conservation Biology 10 12901291 Kendall B E and G A Fox 2002 Variation among individuals and reduced demographic stochasticity Conservation Biology 16109116 Lande R 1988 Genetics and demography in biological conservation Science 24114551460 Lande R 1993 Risks of population extinction from demographic and environmental stochasticity and random catastrophes American Naturalist 142911927 FW662 Lecture 15 PVA 18 Lande R 1995 Mutation and conservation Conservation Biology 9782791 Lomnicki A 1988 Population ecology of individuals Princeton University Press Princeton New Jersey USA 223 pp Ludwig D 1996a Uncertainty and the assessment of extinction probabilities Ecological Applications 6 10671076 Ludwig D 1996b The distribution of population survival times American Naturalist 147506 526 Ludwig D 1999 Is it meaningful to estimate a probability of extinction Ecology 80298310 Mace G M and R Lande 1991 Assessing extinction threats toward a reevaluation of IUCN threatened species categories Conservation Biology 5148157 Mangel M and C Tier 1994 Four facts every conservation biologist should know about persistence Ecology 75607614 Meffe G K and C R Carroll 1994 Principles of conservation biology Sinauer Associates Inc Sunderland Massachusetts USA 600 pp Morris W F and D F Doak Quantitative conservation biology theory and practice of population viability analysis Sinauer Associates Sunderland Massachusetts USA 480pp Murphy D D K E Freas and S B Weiss 1990 An environmentmetapopulation approach to population viability analysis for a threatened invertebrate Conservation Biology 441 Nunney L and K A Campbell 1993 Assessing minimum viable population size demography meets population genetics Trends in Ecology and Evolution 8234239 Pimm S L H L Jones and J M Diamond 1988 On the risk of extinction American Naturalist 132757785 Remmert H ed 1994 Minimum animal populations SpringerVerlag New York New York USA 156 pp Ruggiero L F G D Hayward and J R Squires 1994 Viability analysis in biological evaluations concepts of population viability analysis biological population and ecological scale Conservation Biology 8364372 FW662 Lecture 15 PVA 19 Schneider R R and P Yodzis 1994 Extinction dynamics in the American marten Martes americana Conservation Biology 4 10581068 Seber G A F 1982 Estimation of animal abundance and related parameters 2nd ed Macmillan New York 654 pp Shaffer M L 1981 Minimum population size for species conservation BioScience 31131 134 Shaffer M L 1987 Minimum viable populations coping with uncertainty Pages 6986 in M E Soule editor Viable populations for conservation Cambridge University Press Cambridge England Silva M and J A Downing 1994 Allometric scaling of minimal mammal densities Conservation Biology 8732743 Soule M E 1980 Thresholds for survival maintaining tness and evolutionary potential Pages 151170 In M E Soule and B A Wilcox eds Conservation biology an evolutionaryecological perspective Sinauer Associates Sunderland Massachusetts USA Soule M E 1987 Viable Populations for Conservation Cambridge University Press New York New York USA 189 pp Stacey P B and M Taper 1992 Environmental variation and the persistence of small populations Ecol Applications 21829 Thomas C D 1994 Extinction 39 39 quot and r r 39 quot 39 39tracking by rare species Conservation Biolo y 8373378 Tomiuk J and V Loeschcke 1994 On the application of birthdeath models in conservation biology Conservation Biology 8574 576 Tracy C R and T L George 1992 On the determinants of extinction American Naturalist 139102122 Tracy C R and T L George 1993 Extinction probabilities for British island birds a reply American Naturalist 142 10361037 Urban Hjorth J S 1993 Computer intensive statistical methods Chapman amp Hall London United Kingdom 263 pp FW662 Lecture 15 PVA White G C 2000 Population Viability analysis data requirements and essential analyses Pages 288331 in L Boitani and T K Fuller eds Research Techniques in Animal Ecology Columbia University Press New York New York USA 20

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