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# Wildlife Population Dynamics FW 662

CSU

GPA 3.59

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This 21 page Class Notes was uploaded by Maximilian Lynch on Monday September 21, 2015. The Class Notes belongs to FW 662 at Colorado State University taught by Dana Winkelman in Fall. Since its upload, it has received 86 views. For similar materials see /class/210050/fw-662-colorado-state-university in FISH at Colorado State University.

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Date Created: 09/21/15

FW 662 Lecture 1 Densityindependent population models Text Gotelli 2001 A Primer of Ecology What is a population Krebs 1972 A group of organisms of the same species occupying a particular space at a particular time Cole 1957 A biological unit at the level of ecological integration where it is meaningful to speak of birth rate death rate sex ratios and age structure in describing properties or parameters of the unit Gotelli 1998 A group of plants animals or other organisms all of the same species that live together and reproduce NOAA website httpcorisnoaagovglossaryglossaryilizhtml a group of individuals of the same species living in the same area at the same time and sharing a common gene pool a group of potentially interbreeding organisms in a geographic area De nitions taken together point to some unique aspects of populations 1 Abundance or Density 2 Have boundaries 3 Change over time 4 Composition sex and age 5 Distribution pattern scale What is quotpopulation dynamicsquot It s an area of investigation between the elds of population biology and population mathematics Population Biology Population Dynamics Population 39 quot quot Little or no concern for Balance of biology and Little or no concern about mathematical mathematics biological reality representation Some questions we would like to be able to answer after this course 1 Maximum number of adult females that we can harvest sustainably from a mule deer population 2 Number of individuals needed in a reintroduction e g wolves into Yellowstone to be fairly certain that the population will persist 100 years 3 Given data on survival and reproduction what are the chances that an endangered population will persist 100 years 4 How large of a marine reserve is required to prevent the extinction of a harvested fish species FW 552 Lecture 1 Densityindependent population models Some of the topics Lhalwe Will cWel39 include L 39 39 lniwi r mmmh ansstructured populations stagestructured populations additive vs compensatory mortality stochasticity demographic environmental individual heterogeneity genetic sampling chaos metapopulations predation competition herhivory evolution natural selection genetic ocial 19m 39 39 viahili 39 population Dynamics is concerned with changes in the density or numbers of organisms and the processes that cause these changes Nu gt I C a D L O a O C U E 3 Nt l Q lt Time Pnnula on relationships 39 39 39 L Some thoughts on modeling Hence 39 39 39 t w 1966 bulldmgmpnpulahnnblnlngy Am Sci 54 4217431 39 39 39 quot outthat that models L modeling is essentially a tradeoffbetween Z Realism 3 Precision FW 662 Lecture 1 e Densttyundependent popu1atton models The usefulness of any panxcularmodel depends on the modeler39s goa1s To desenbe der w Wuhams et a1 2002 Chapter 7 Desenbe model usefulness Wth the following eoneeptua1 model Empmcat Strenglh Biolugtcal Understanding strength 1 models tneotpotattng good btologtea1 understanding and supported by data W IV model based pnmanly on speeulauon and poorly supported by data Advantages tn mathematical mndels quztinns can he manipll lat d Thxs leads to embthty beyond what is posstble wtth a eoneeptua1 model 2 Equatinnsreqlure exp it statements nfnlles and vanahles t eads to more pneetse statements or hypotheses 3 Equatinns cznresll t in Emergent pmpem39es Sue propemes are durteult onmpossxble to preddct usmg nonrmathemaucal teasontng FW 662 Lecture 1 Densityindependent population models Adults ppr0portion of eggs that survive Beggsl adult The number of adults N is some function F NtFtl the number alive at time t is some function ofthe number alive at time tl So what is F F pB Then NtpBNtl Then N pBWH or NH PBNt A mathematical aside change difference and ratio equations 39 39 39 Dirrerence Three imp unant unes change Equz39im MA fNt Difference Equatiun NH N fNr MA Ratququatiun M f N FW 662 Lecture 1 Densityindependent population models Geometric Population Growth In population dynamics we are generally concerned with four fundamental rates Births B Deaths D Immigration I and Emigration E We refer to models using these parameters as BIDE models Change in abundance over time is modeled as a function of how many individuals are born die leave or enter the population NH1 Nt Bt Dt 12 Et Initially we will consider closed populations ie n0 immigration 0r emigration and nonoverlapping generations or change is measured in discrete intervals Biologically this means pulsed reproduction and nite survival N21 N2Bz Dz we need to convert the raw Birth and Death rate to per capita rates per individual because per capita rates are easier to estimate bt and dDvt N21 Nzszz dzNz Substituting Now we assume that per capita birth and death rates are constant over time NH1 N2 bNt dNt Simplifying this equation NM N b CAN The term bd is the geometric rate of increase and is given its own symbol R NH1 N2 RN2 NH1 N t RN 2 a difference equation AN RN AN ZR N The parameter R represents the per capita rate of change in the size of the population Lets go back to N21 N2 RN and factor out N1 FW 662 Lecture 1 Densityindependent population models NHI RNl In population ecology 1R is given its own symbol lamda a change equation this is called a Taylor Series NZMJXM4WNHNXM Mwm using this equation there are three possibilities for population growth Growth Geometric Lamda can also be expressed as ratio a ratio equation 5 Hence we can calculate lamda without knowing the per capita birth and death rates however in actuality it will probably be important to estimate b and d FW 662 Lecture 1 Densityindependent population models Exponential Population Growth Now let s suppose that we have continuous population growth or change is measured over a very small interval Biologically this means continuous reproduction and instantaneous survival rates This requires a differential equation bN dN dt Let b dr rN dt r is called the instantaneous or intrinsic rate of increase With integration we can get a predictive equation for N Put in terms of N rdt N from calculus d lnx x so d lnN rdt Z Z Integrate Id lnN Irdt 0 0 lnNt lnNO rt take exponential of both sides 1 N 71 N 0 e n 06 n Z en recall that em a and 6 1quot 1 a N er N Solve for Nt and we get a continuous model that predicts population size N Noequot t FW 662 Lecture 1 Densityindependent population models Using this equation there are three possibilities for population growth We have derived two predictive equations for population growth Discrete Continuous Density Independent NI INO NZ 2 Noe Comparing the two models A e or 1nd r if r is small then we can use a Taylor series to nd the approximate relationship r r2 r3 6 lr 2 3 A 5 1 r The two models diverge as r gets large and as the number of generations t increases Consider the following example with R02r FW 662 Lecture 1 Densityindependent population models Rr 02 N t N 0 4 t l N t N 0 e r l 0 1000 1000 1 1200 1221 2 1440 1492 3 1728 1822 4 2074 2226 5 2488 2718 6 2986 3320 7 3583 4055 8 4300 4953 9 5160 6050 10 6192 7389 The distinction between R or 7 and r is important biologically ie pulse versus quot r J quot 39 versus nite survival rates There is confusion in the literature between r R and 7 so be careful to understand the author s de nition FW 662 Lecture 1 Densityindependent population models Some Useful References Begon M J L Harper and C R Townsend 1996 Ecology Individuals populations and communities 3rd ed Blackwell Scientific Ltd Cambridge Mass 1068pp Caughley G 1977 Analysis of vertebrate populations John Wiley and Sons New York pp 17 Cushing D H 1981 Fisheries biology a study of population dynamics University of Wisconsin Press Madison on reserve at library Donovan TM and Welden CW Spreadsheet exercises in Ecology and Evolution Sinaueur Assoc MA Hastings A 1996 Population Biology concepts and models SpringerVerlag Inc New York Williams BK JD Nichols MJ Conroy 2002 Analysis and management of animal populations Academic Press New York FW 662 7 Densitydependent population models I In the previous lecture we 391 J densit 39 J r r r models that assumed that birth and death rates were constant and not a function of population size Longterm density independent population growth is unlikely and an unrealistic assumption Birth and death rates are more likely a function of population density or abundance Density Dependence births are a decreasing function of density bN and deaths are an increasing function of density dN bord This results in population growth being a declining function of N Population Growth Hence population growth will be zero at some population size This point is usually referred to as K or carrying capacity but let s develop the model first considering the explicit functions for birth and death The approach is described in Donovan and Weldon 2002 but I have modified it to match the notation in Gotelli 1998 FW 662 7 Densitydependent population models We need two new terms to account for changes in per capita birth and death rates a the amount by which the per capita birth rate changes in response to an addition of one individual to the population c ditto for death rate Our density independent discrete model was a difference equation expressed as NH1 N2 bNt dNt We replace b and d the density independent birth rate and death rate with b aNt and dcNt Now our new density dependent model looks like NM N b aN N d cN N Population growth rate is not easy to Visualize from this equation But you can see that initially the population will grow geometrically because Nt is small but as Nt grows a and c have a greater in uence on the population The question is will the population decrease in size Increase Or stabilize Use a common tactic and assume that there is an equilibrium ie a point where the population size isn t changing N2 NH1 NW NW NW b czqu m d chq m Subtract Neq from both sides and add d chq M b czqu M d cN2q DiVide by Neq b CINE d cN2q This tells us that the population is at equilibrium when per capita birth and death rates are equal THIS MAKES SENSE FW 662 7 Densitydependent population models Now we can solve the equation for Neq Rearrange the equation by subtracting d and adding aN b d chq 61qu Factor out Neq b dacN2q DiVide by ac b d 2 NM 2 K a c Using this equation you can calculate K if you know b d and the two factors that account for density dependence Density Dependence discrete logistic Typically K is speci ed and not calculated from the birth and death rates Let s start with our geometric model AN RN The existence of a carrying capacity K suggests that the population cannot exceed this level N ANRN1 K N N21N2RN21 N NM N RN1 K FW 662 7 Densitydependent population models The logistic model results in the following population dynamics When Nt is small then geometric growth and when NtK then population growth is zero Pop Size We can also look at the change in population size as a function of population size Initially the population is growing quickly but then decline to zero Delta Nt Delta N Looking at the per capita rate of change the population begins growing at R and declines to zero at K From this graph you can see that the density dependence is linear DellaIllNl FW 662 7 Densitydependent population models Density Dependence continuous logistic rN KN dt K This equation predicts the rate of change in population size and you need to derive a predictive equation WHICH IS This model results in the same dynamics that were demonstrated for the discrete version of the model However the discrete version of the model is capable of demonstrating a wider variety of dynamics than the continuous version which we will explore later We have derived two predictive equations for exponential population growth and two equations for density dependent population growth Discrete Continuous I rt Dens1ty Independent NI y N0 N Z N Oe K K N NI 2 Density Dependent NH1 N2 RN 1 N0 6 N0 FW 662 7 Densitydependent population models Other models with Density Dependence PLEASE PRINT AND READ GARY S LECTURE NOTES 57 Many possibilities exist for describing density dependence and there are several other models that have been developed The Ricker equation Ricker was a fishery biologist interested in predicting recruitment to shery stocks and developed the following density dependent equation Note that density dependence in this equation is not linear and becomes stronger at higher densities due to the exponential function N HI NteR0lj The Hassel Equation sztae0 1 Nt1 KMamp FW 662 7 Densitydependent population models Comparing population growth among the logistic Ricker and Hassel models for R075 and K100 o 15 2o 25 Time t Comparing the form of density dependence among the logistic Ricker and Hassel models x xi delta Nt I Nt 9000 omnmoo 0 20 40 8 80 100 120 Comparing the per capita growth rate among the logistic Ricker and Hassel models 25 20 delta Nt 8 I m 0 20 40 80 100 120 60 Mt FW 662 7 Densitydependent population models Maximum Sustainable Yield MSY I will develop a model based on the logistic difference discrete equation and follow ther derivation in Williams et al 2002 Start with our discrete logistic difference equation N NM N RN1 K and include a term for total Harvest Ht N NH1 N2 RNtl Ht The population is at equilibrium NtlNt when N RN 2 I J H t K and the per capita harvest rate h H A hl 24 K A given per capita harvest rate ht corresponds to a speci c equilibrium population size that can sustain it and can be seen by rearranging the above equation in terms of N MilQ This equation indicates that the population can be sustained in equilibrium for any value h that is less that the population growth rate R The question is What value corresponds to the largest sustainable harvest R 2R FW 662 7 Densitydependent population models Set this equal to 0 and solve for N which results in NI Substituting into our equation for equilibrium harvest RN1 N H K Ht RKAH Then the per capita harvest h is h HZW W MSY FW 662 7 Densitydependent population models 0 0 5 0 0 4 0 0 3 w Y 0 S O M 2 0 0 1 0 119876543210 4 2 o 8 6 4 2 0 10000000000 1 1 1 0 000000000 32 38 3232 303 300 400 500 NM 200 100 FW 662 7 Densitydependent population models Gotelli N 1998 A primer of ecology Sinauer Associates Inc New York Hassel MP Densitydependence in singlespecies populations The Journal of Animal Ecology 44283295 Ricker W E 1954 Stock and recruitment Journal Fisheries Research Board of Canada 1 162465 1 Ricker W E 1975 Computation and interpretation of biological statistics of sh populations Fisheries Research Board of Canada Bulletin 191 Ottawa Canada Williams BK JD Nichols MJ Conroy 2002 Analysis and management of animal populations Academic Press New York

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