Fish and Wildlife Population Ecology
Fish and Wildlife Population Ecology WLF 448
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Lotkav Model dHdt bHP Hnumberofre r prey population growth rate b attack rate P number of predators dPdt CHP quot l39 c predator population growth rate due to predation k rate of predator decline in absence of prey Lotka w I V Model dHdtlt0 dPdtlt0 N dHdtlt O dPdtgt0 N Vquotquot rb l u I K dHdtgt0 dHdtgt0 dPdtlt0 kc dPdtgt0 Modified Lotkav Model dH dt r H 1HK aHP 1aHh dPdt cP 1PJH k P H number of prey r prey population growth rate b attack rate P number of predators c predator population gro m ue Iredation k rate of predator decline in absence of prey J prey density required to support 1 predator per area Stability Tanner 1975 Ecology 56855 Explored features ofthis model to find general properties particularly model stability Does the quotcritical point where predator and re isoclines cross roduce a stable equilibrium quotfocus point limit cycle unstable predator growth prey gwwth ICILCD sr note c s Stable focus when the critical point falls LU the right of the prey zero isocline peak for all values of sr Tanner1975 ma 5A a at Tanner1975 When the critical 38 point falls to the left of the prey LEIU isocline peak 1 stable focus if sr is large i Mme H Tanner1975 36 by When the critical 39 point falls to the in left of the prey LEIU 391 isocline peak 2 limit cycle if sr small T small P HE When the critical point falls to the left of the prey LEIU isocline peak 3 unstable focus if sr small and K is ver larve extinction no coexistence Tanner1975 So a quot Tanner1975 What if predator is limited a resource that is independent of both predators and prey such as nest sites or space mu than prey or predator numbers gtP Once again since the critical point falls U the right of the prey zero isocline peak a stable focus results for all values of sr Tanner 1975 Tanner 1975 Again since the critical point falls to the right ofthe prey zero isocline peak a stable results for all values of sr V w I 1 l Tanner 1975 When the predator and prey zero isoclines cross three time two and one unstable saddle points are created Population can jump from one to the other depending on starting point and other model constants or Unstable focus Stable focus a Stable focus Tanner 1975 The prey population can get stuck at very low density unless I drop called a predator Unstable focus Stable focus up 0 Stable focus Predator Pitquot Tanner1975 0 Complex model behavior nearly any outcome So what Is this useful Tanner1975 Complex model behavior nearly any outcome So what Is this useful Tanner reflected on the general patterns from models Hypothesized that stable prey species were either strongly self limited eg by territoriality or the prey population growth rate was less than that of the predator How would you test Tanner1975 Hypothesized hu stWIe p species n either stroneg self limited eg by territoriality or the prey population grow rate was less than that of the predator Prey growtn IaLc appcalcu IIISIICI Sl lt J IUI sparrow hawk house sparrow and Mink muskrat And both prey species thought to be self limited s arrows food or breedin39 sites muskrats territories Tanner1975 Hypothesized hu stWIe p species n either stroneg self limited eg by territoriality or the prey population grow rate was less than that of the predator Prey growt ICILC appca cu sma W 1 u Lynxsnowshoe hare Ha e Iylll JIIUW Tanner1975 Hypothesized hu sthe py species n either strongly self limited eg by territoriality or the prey population grow rate was less than that of the predator Prey growth atc appeared owersr gt 1 u several prey species with weak selfregulation Mlin ml er Wolf moose caribou WT deer white sheep Model wumptions No time lags No prey refuges Predator searching constant not w w external factors NO differenees III plcy susceptibility Opt waging Theory Ho does a predator choose which prey hunt and for how long Theor develo ed to identif the o timal choices based on profitability of prey items or foraging patches where profitability energy handling time The optimal diet or foraging patches are those maximizing profitability Perfect match unlikely because animals must explore choices to learn profitabilities and profitabilities change through time Model wumptions No time lags No prey refuges Predator searching constant not w w external factors No differences m plcy susceptibility Prey switching and switching of habitats Predators switching to another prey at low prey density essentially creates a refuge theoretically increasing stability Evidence Hanski et al 1993 and Turchin and Hanski VUIU pOpUqulull uynallllbd In IIOItIIbIII I ulvrlb Few relatively specialized predators in northern populations Numerical response to increased voles With a time lag More and more generalist predators in southern populations Relatively constant population size Rapid behavioral response functional response to increasing vole densities m al 1997 0 Modified Tanner model Seasonality Stochasticity Parameterized with independent data Tested the hypothesis that popUIatIon Stapllity depends on type of predators Model Predictions A GD Obsened dynamics ILPISJARVI VOIL39 lilplllJllUll iymmm t mlwm lmannmAllianmylauedimugidm m 0 mm an lwwnl mm mm milivcnn clminn lwm Northern Populations Relatively specialized predators Few alternative prey Southern Populations Relatively generalized predators More diverse prey uu mile mm mt mm m cm lrnjrcmry Can predators limit pop to Connolly 1978 WV of ungulate studies 31 studies provided evidence of predator regulation 27 did not 39 Cote and Sutherland val IVIeLd analysis of bird predator removals showed increased hatching bULLcSS and larger post breeding population size and breeding population size in next season Introduced predators Freshwater syn appear N be much more sensitive to introduced predators than prey in terrestrial syn on wcw but not islaw Cox and Lima 2006 argued terrestrial prey are seldom nai39ve LU predators because of hist0ncal biotic exchanges among continents Marine prey have also expenenceu biotic excnanges 0 In contrast FW habitats have high heterogeneity in predator regime and lower dispersal rates promoting naivet in prey Indirect effects and the quotecology of fear Lima 1998 I quuuvru uuu rilby 0 Also stroneg affect prey behavior 0 When obtaining food is dangerous altered behavior may affect prey foraging rates growth rates survival and population growth ratesin other words fear may reduce indirectly fitness quotindirectquot or quotsublethalquot effects 0 behavioral trophic cascade Yell to rthern winter elk WHO cumin SMngs Slraamgage u rm smug slla Innlllssll esll J Cugke any on l n 1 mm 7 m Nolllvem wlnm Range Rwers nm 7 avk some Lucmmn armay was wlhm he nmllveru Yellmulane elk vmlerumgc Fm 2 Yellowstone a established in 187 Policy of ungulate protection begun in 1886 when US Cavalry began managing park NPS continued policy when it assumed control in 1918 Wolves hunted and poisoned in and out of park during this period last recorded in 19205 Simultaneously administrators became concerned about overgrazing Removal program 10000 elk in 19305 to 3000 4000 in mid 19605 Yellowstone Control efforts ended in 1968 period of quotnatural regUIatIon followed Herd increased to average 10350 in 19705 15550 in 19805 and 16570 in 19905 Wolf reintroduction winter 1995 6 2002 northern range population 78 Elk constitute 83 of annual wolf diet Elk grazing controls height of woody browse aspen willow cottonwood Wolves alter elk foraging location and rates Cottonwoods n v a 2000 Beschta 2005 Ecology 86391 Cottonwoods H v a 2000 h Beschta 2005 Ecology 862391 Beschta 2005 Ecology 86391 Climatic factors Elk refuge Elk refuge Elk range Elk Range Bison Ranch Number ol Irees We Wam Pugml mm gt a La Duke Sprmg a Lamar Rlver I mm D mummy V w culllnu if 7 q q a a rm 3 a q o s amps v agt e iamp no an El g S a W NV g lt2 NW e 35 lt4 3 4 a 39 Establishment dares decades m WWW m 7 Flequeuc manldales luruees c n 39e study vres m llve l nl luvmnad mum 7 breml lmgluklbl l elk mm Trophic WWW Does this cyr a classical uuuuuu ca Or a behavioral trophic cascade 39 nemoval program 10000 elk in 19305 to 30004000 in mid 19605 Herd increased to average 10350 in 19705 15550 in 19805 and 16570 in 19905 Beyer et al 2007 Compared willow growth rates from before and after the reintroduction 20 w 2 gt 4 ng area mmZ f 39 7 F T T Ring area mm t H PH r w wen wee 15m 1995 was anon H4 4 nnlul nng was man rm A SUn Mnlhrz mum us s gmuvmm mm Miami m 2001 M n 4nd ugh umphng sues r p on Ycllnwmnc s as L nunhu mg m mam hm mum m nm mmcr m mylvb mt mm on m no mm g mm mm Mam mum mmmdm Beyer H 2007 Tm 1 mm h mm 1 MC smghh u rm mg mp tanduhk model m1 mu mamm mm rclumg mum m mum mu ling m 0mm bunmm x gEu39Hmm an Vcllmwmnc s ngu mumquot m Mndcl dmnpnun LL MC ANC n IPA wou 71055 2 21234 on n s um wsamwou 710545 21291 us 015 L VN 4 L1 71057 21127 41 mm mm pusx wsuw wou 4055 x m n 52 u an anv PRLCIPA 7 mm a 213x 5 m z u an PDSLL K 7mm 21722 42x mm Snw gu uu39mm Lm pm on m prw AS 000 wa pump 4 5 o as anV m 71 002 NPIW M s a an vwa tLK 4 45 146 2 172 mm mmquot wmhlm mm mm L LhV mm 4m WSHLD mm mm Norm mm mm NHW mm mmul plL LlplLuiLm LPRLCAPA mm gmwing vmsnn prudplmnnn Mmrmglm mumps mc mum Palmer drought many mch wus a huury umhlc mpmning my pmiumc m mum an m dupe wou m1 m nurmm mngc Lu m Muth mu my hmwccn ymrs hm nm bum mus mmquot A ymr mu mudch mm my gnu m mpmm I m mg mp mmick x u mfcmm39c moduli mcludc my Inghm mm model mm m wou mr39uhk mow m m mm man rmch um mm m ELK mml c hm nm the wow wrubk Vulka Abbrcvmmn5 m mmle m Mumn Elk population iz 6 g m u 2 1 5 u i i i i i i i i i i 1986 1990 i994 1995 2002 Icn0rlhcm i t 2 i5 poplilnliontoumcimlid Armin unl mmcr mi of Yellumtunc NMimLil pm USA rupnnd to m Dnccm u n kpun vink i ithlhcctmmmu l g N ununlsmukplium mg mum of ms mi 1995 u in mm mm mmium mt mum rm 1mm 1990 mum undmmmm ii iii Ami Gurmll 2005b counts Adjll lul rm igm ihiiiiy Upcn inm m h1 uwinlu5utm csliimlul h Coughmmir and Singer was Herd size declining at 45 year since reintroduction but not a good predictor Behavioral Trophic Cascade Changes in elk distribution or feeding habits Byer et al 2007 Ecol Appln 17 1563 Beschta and Ripple 2006 Wilom cottonwood aspen are all riparu species Can WOIVes afreu stream riparian vegetation and stream channel morphology 1949 1924 Figure I Pham chronusequence o m GaHaUn Rwer and uadmam Khuwmg the sum 01 npmn wwHow mmmumuex amng mach a m A summer H mm a summer m 949 c be spvmg u wanna D summer a ZDDEJH ghhmrg39 LE uss cl mm wax I muggy A an H n H N h dump wen puman M n w mm m u v n H rwhr n vr M m w n 39ure4 Nr rv nhhe reach a photograph represent same that have begun to recover me we ve rammed m m ham m the mmewgm mum r my dun m Flood uency g 7 Reach SE 60 g x 50 quot Reacha g 40 021 0 xx 5 30 quot 1 R MA 6 eat 20 11 amp 39 to n39 10 mu Return Period yrs Figure 5 Dwxchargehequency m anonsh p dashed hues m endwes A a m 5 Wow regmna equanons m men and m H mm at am study eazh we shaded m encompasses the We m remn permds or enzhA iontm Mode Conceptual Wolve Present Wolves Absem meme 9 prey 7 Plams fun Channel Murvho nny ltee a Hynmxogle Canneclivlw Weememenemmeee We meme U 4 Elk numbers and panama or hamlm use veeuu m ewemaemem bmwsmn m manequot vege a qn hmwsn species m nuanan area m eexemem nmw and yepmdm n eve s memes be owgwund hmmass wnmbule m In em roughness and bank seem Wmmy mg Ge alm awe has a revexwewmue meandehng mglelhlead meme m Yang M nemean W5 new mgmu m M has m eenmu Enws invraxxmztsw wan2 yeeyemnemege m mn unmmn wuh lugh waze we wow e m mews cnnmhnns malauslam Hpumm mammmmunmes moreasee em numbers snmcr ummneaed namm use mm m heavy hmwsmg e4 annan vegmamm Reduced ahm and eeuewgmune mamas niwandy brawn pnalli ausns ms at hydrauHc roughness and mm suenglh v Acne eralen nenx eromnn mm m wmesureed channn adlummcnls mcludv g Mammy mm and avulsmm sums evewmenea reachgs agqmde 0 OccurJenna emenkmu news qenera ly exceed u yams m mnguneuen wxlh xewenn er gromvdruler Eva s cause sad mm m mun um rm may seem npanen mm cummunme Figure 7 Summary OV39Luprduwn39 mth asmdes sohd arrows and hymogeemorpmc pmcEsxes dashed armwx uncepma Hp n uence man suctesxmn and s and mtemztmm 5 ee her we not mduded eg Bawm et aL 2005 kuMng such 1mm as envuaumenm nHymg capanty k annexe Life Cycle of Callibaetis ferrugineus hageni Imago Subimago Hours Days Hours Days Stoneflies amp trout cues decrease grazing on algae behavioral trophic cascade 800 b Diatom 600 No Stoneflies 400 Stoneflies O u 1 absent present Fish odour Baetis size at emergence in natural populations Summer Femuies r13 mm P uumr White shless stream Black trout stream at More r toioitiie t39wetuitir i eat to 9 Mules BMW FUIUI74 Peckarsky et al 2001 r u or sit MM eteiborie oio i rio ruie 9 9 Siteyears Whole w manipulation 15 I Fish odnr ET Fishless control B E m 1WD m m E E 7 D g 05 E a D Female Male FIG i Dry mass mean i E SE 01quot mature black wing padj female and male Baeris larvae was inwer in streams with bmnk mm chemicals added solid bars compared to comm streams with uni shleas water added open bars Data are for the summer generation 1999 Indirect effects of trout cues gt direct effects of trout mortality on fitness A demogup ouc suggcs tu cwg trout mortality would increase fitness by 388 A natural puyumJun 1 vs 7w 0 However removing the indirect negative enects of trout on growth would increase fitness by 1140 A4264 McPeek and Peckarsky 1998 v mayflies Could w mayy fitness indirectly Fish Projects Flow USGS Stream temperature Not much available Could use proxy measures Ranks high water vs low water warm vs cool Population Models So far we ve focused on Estimation techniques Characteristics of populations Introduction to population growth models Unlimited resources density indpendent growth Limited resources density dependent growth Population Models Forecast future conditions Sustainabe yield Population viability analysis Trajectory of population size for invasive species Hindcast to explore potential mechanisms Population Growth Examples Human population mommaquot m Iaime American shad Columbia River annnnnn 8 E a mum 4 innnnnn n i n 195 195 15m war war mun 2m Population Growth a simple case Constant environment Unlimited resources All animals are the same Population Growth a simple case Change in numbers AN AN Births Deaths Immigrants Emigrants Ignore immigrants and emigrants Assume closed population or assume I E or combine B and D E AN B D Population Growth a simple Population Growth a simple case case ANN1 N0 ANN1 N0 N1 N0 Births Deaths N1 N0 Births Deaths 39 NlNORBNORDNO 39 NlNORBNORDNO 39 N1NORBNORDNO 39 N1N01RBRD Population Growth a simple Population Growth a simple case case ANN1 N0 ANN1 N0 N1 N0 Births Deaths N1 N0 Births Deaths Ni N0RBN0 RDNO Ni N0RBN0 RDNO N1N0RBN0 RDN0 N1N0RBN0 RDN0 INlN01RBRD 39N1N01RBRD x1RB RD x1RB RD N1N07i N1N07i Note A N N0 Population Growth a simple Population Growth a simple case case Let s project Let s project N2N i N2N i N2N0xx N2NoxZ Population Growth a simple case Let s project N2 N1 N2 N0 7 9 N2No AZ N3 N0 M9 N4 N0 XXX etc In general NL N0 N N NON Population Growth a simple case This treatment of growth rate A is very simple and intuitive A 106 6 increase per year Assumptions Birth rate is constant Death rate is constant It treats all members of the population as equal or it assumes a stable age distribution Reasonable for some populations Nonoverlapping generations insects annual plants A can also be estimated from agespecific birth and death rates Note only describes the population size pertime interval What if species does not have seasonal reproduction at if we want compare species with different intervals of population change the tortoise vs the hare A is mathematically cumbersome in some instances Use calculus based continuous time analog dNdt r N N NON 1n NL In N0 1n Mt 1nNL1nN0rt yabx NLN0A NFN N 0 391nNt1nNo1nAt 1nNL1nN01nAt 391nNi1nNort 391nNL1nNOrt 39yabx 39yabx 1 A 2 FInIte and Instantaneous Rates A 113 year r lambda 051033 as What Is the daily rate 022314 08 005129 095 0 1 004879 105 m N 0182322 12 0336472 14 0693147 2 2302585 10 1 Finite and Instantaneous Rates Finite and Instantaneous Rates A 113 year Instantaneous rates can easily be What is the daily rate subdivided but finite rates can t Instantaneous rates can easily be A 113 year subdivided but finite rates can t r 012year A 11312 0094 month 906 001m0nth decrease month 0000329day A e 1000329 day 003 day Finite and Instantaneous Rates Finite and Instantaneous Rates Finite survival rate Finite survival rates are multiplicative S NI NO Instantaneous mortality rates are additive Instantaneous mortality rate Zvveek 7 Zdailv In S S e Z Unlimited Growth Assumptions Changing environment b d is constant implies constant environment and unlimited resources 39 POPUIailon Change 395 In the real World 395 All members ofthe population are equal or dynam39c b and d Change population has a stable age distribution 39 ObserVed Change 395 caused byi Reasonable for some populations Real Challg s Process error Nonovertapping generations insects annual 39 determ39msnc causes lants stochastic process error if t d tk b 39 Nonetheless Slmplsexponentlal S Zimograpm gromhmodfls prov39de good Samping observation error predictions In many cases eg collared dove in England We can Incorporate Into models Stochastic population growth Stochastic population growth Mills 2007 Figure 55 mummsmmag Stochastic population growth Stochastic population growth I lmhdz N LGhiz N El 1EE a 1EE I lurk12 N zrhiz N unidz M 1 12 12E 12 12E U 1 39 1 39 1 2 12 144 14 158 1 1 12m 1 12m 1 12m 3 12 m 1 EB 2 12 1 1a 151 1a 192 A 12 m M 185 3 12 113 1 151 m 15o o 1 2m 11 115 na 131 g 1 g 1 323 5 1 2w 13 2m 15 217 7 12 358 13 w a 12 259 11 2m n5 111 a 2 an 2 i 1 22 1 1 1 2 WEE l 2 l 2 mvage 1 2 1 2 1 2 Smng Emu U131 SdDw EIEIEIEI U131 U312 Stochastic population growth Unlimited growth summary 1 mm 1 mm 1 mm M n 1m 7 1m 7 1m Unrealistic longterm assumptions 1 1 12m 1 12m 1 12m 1 13 11 122 g 13 Finite and instantaneous forms each have 2 1 2 1 2i 1 2 advantages 5 12 259 11 2m n5 1m 7 12 358 13 a 15 m StochastICIty affects ability to accurately a 12 m 12 m 12 335 predict future conditions Nimman 12 12 12 Side mm W m Prowdes accurate predictions in many Gammaquot 12m 11m 11m cases m short time intervals A6 A1 A2 A3 At invadingcoloni2ing populations Hm quot dynamics Sex Ratio Introduction I Must be careful in examining sex ratios l Males more obvious plumage often I In breeding season males more obvious beha viorally l Sexes may migrate separately Terms I Ratios commonly expressed l Males39femaes to make 100 total 5050 I Males 100 females I Percent of total which are male or female I Less commony as femalesmale Terms I Ernst Mayr 1939 suggested a classi cation based on age I Primary sex ratio at conception I B Secondary sex ratio atbirth l C Tertiary sex ratio at later speci ed age Mammalian and Avian Patterns I A Basic tendency l Mammals Sex ratio shifts toward more females in older age classes I Birds Opposite shifts toward males predominating in older are groups Mammalian and Avian Patterns I B Why What are reasons Two types of expanations l 1 Internal Sexlinked lethal factor I 2 Environmentaly induced mortality related to behavior and life history of sexes Examples Humans I Why do women live longer than n l Holden C 1987 Science 238158160 I Women outlive men by a margin of 4 to 10 years throughout the industrialized world Weaker Sex I Now that females are no longer being felled by childbirth it has become clear that they enjoy an advantage in survivarates l James V Neal Univ Washington We really are the weaker sex biologicaly less fit than females at every step of the way By Age I At conception 115 males to 100 emales I At birth 105 to 100 l Male excess In spontaneous abortions miscarriages and stillbirths prior to birth and higher neonatal and infant mortality I At 30 sex ratio is equal I By 65 84 females amp 70 males still alive Male to female death ratio I 39 to 1 for homicide in Alameda County Calif I Also higher for lung cancer suicide pulmonary disease accidents cirhosis and heart disease 2 to 1 Y chromosome I Maleness seems to carry intrinsic risk I Study ofAmish families with and withoutong arm of Y chromosome I In families with women died In mid70 s and men 5 or 6 years earlier I In families without women died atave age of 77 while 14 men died at82 Hormones I Males and females have equivalent cholesterol levels until puberty l Males suffer an exponential rise In heart disease In their 40 s but female rise does not start till 50 s after menopause l Animal and human studies show that estrogen protects against heart disease by lowering levels of lowdensity lipoproteins LDLs bad cholesterol and keeping highdensity lipoproteins HDLs good cholesterol up Hormones I Androgen lowers HDL and raises LDL l Hormones also a ect the immune system I Animal and human studies show that females have greater immune responses than males I This is associated with higher female antibody responses and tumor resistance I but also higher female rates ofautoimmune diseases arthritis lupus Examples in Birds I Table of Sex Ratio Changes Examples in Birds I Sex ratio in young close to 5050 I In adults it increases towards males I Usual rationale for higher mortality in females is stresses and hazards of nesting Examples in Birds I Pheasants in VWsconsin l Wagner 1957 found I late summer mortality higher In hens when nesting pushed later by poor spring I Grouse in northern VWsconsin I Dorney and Kabot 1960 saw I year to year sex ratio changes a ected by spring weather poor nesting weather killed the females Stress I Relating female losses to stress factors I Stress is somewhat intangible and dif cult to measure I Can look at physiological condition Physiological Condition I Peterson Oldsquaws Physiological Condition I Most birds at peak physiological condition just prior to breeding season I After laying rst egg females condition I Extent of decline depends on severity of breeding season I ie number of clutches number of eggs Energy I Energy stored as I glycogen I lipids I protein I Dun39ng laying and incubation these stores are depleted I Gallinaceous birds hens start out with relatively little fat Gallinaceous I Large clutch or renesting leads to decline in condition I In chickens layers at end of season are skin and bones breasts are hatchet shaped I Deplete protein reserves I Moult follows placing more stress on them Canada Geese I Hanson and Raveling saw I No food during egg laying and most of incubation I Need large fatreserves for eggs and own energy requirements I Every day that spring is late bird burns up enough energy to lay one egg I So ave clutch size declines one egg Predation I Females nesting on ground or low in vegetation and undertaking all of the incubation typically may suffer more predation than males I Keith 1961 studying waterfowl during nesting found I 2 loss of drakes I 8 loss of hens mostly due to predation while nesting Intraspecific I Perhaps females lose out In competItIon for food I S Dakota aftera hard winter 75 of phesants found dead were females though they only made up 57 ofpopuation I On Protection Island sex ratio changed from 5050 after introduction to 6040 at high pheasant densities Examples from I Higher mortality rates in males may be due to greater activity I a Larger home ranges I b Reproductive behavior ungulates rut in late fall and go into win ter in poor condition I c Physiology testosterone in bulls inhibits fat deposition Richardson s Ground Squirrels I See table in class notes p 54 I Juveniles 5050 sex ratio I 11 months 3070 I juvenile males hibernate 1 mo later and emerge 2 wks eary periods of no food and high predation I 12 months 1189 I Males driven out by females and dominant males to die in poorhabitat Ungulates llmbalance toward females accentuated when food shortages occur on overutilized range lEk Cowan 1950 Flook 1970 I Reindeer Klein 1968 Snowshoe Hare I No change in sex ratios apparent Sex Ratios in Fish I Vary considerably but close to 5050 in most Nickolsky 1963 I In freshwater shes most studies show more males in young of the year while very strong preponderance of females in older sh Sex Ratios in Fish I Males produce many more sperm than females eggs I If too biased fertilization declines I Trout on spawning grounds Significance of Sex nu u I Dependent on mating system I Component of ef cient herd management in mammals Patterns in Mammals I Younger females produce more male offspring I First offspring at any age is more likely to be male Natality Patterns I 1 Youngest reproductive age classes have lower birth rate than adults I 2 In less favorable environmental conditions reproductive rates decline most in younger age classes Natality Patterns I 3 In sh reproductive success is extremely variable Stochastic Population Growth Stochastic Population Growth 9 nrr l lr39nnnki emr39 River SparHers Counted jun nun Chmuuk 19E mussmsmm vzmmsmmamamaawmaaa l L ear I Stochastic Population u wt I Dennis Munholand Scott 1991 I Stable but variable mean rate of change I Constant long term trend Trend ILet ri In NiNH I Ti 39 ti1 I Trend Mean change time I Estimate of trend Mean ri Iu EriEri ln nqnutq t0 I In 83839012 27 I 01422 Variance of rate of change o2 Erawin rrzrq 1 l 0645 Predicting Future n l 4 I39U39JUIGIIUII Predicting Future PUIJUIIGtIUquot I We cannot predict exactly what the population size will be I But future population sizes are not all equally likely I We must describe them with a distribution Predicting Future n l 4 I39U39JUIGUUII robable opurauon rze tamngar Predicting Future Pupur39aiiun I Mean NoeXp u 05 o2t I Median Noe M Predicting Future n l 4 I39U39JUIGIIUII 39 robab e opmatron rze mg a k m Probability of Extinction I 10 forult 00 I p I exp ZuXdGz forugt00 I where Xd In NqNe I Ne Extinction Threshold I X d In 84030 Time to extinction l Conditionalmean time to extinction I T X d I r I I In 840300142 234 yrs Is there no hope 10 foru lt 00 I p I exp ZuXdGz rorp gt 00 I what if w ed water first for sh and ocean conditions cooperated so that trend was flat IFO 001 I then pextinction I for W003 pexlinclion 03975 39forrLFOlo pexlinclion 00002 Estimation I You can calculate the values for this model of stochastic population growth on a calculator by calculating the natural log of lambda at each time period and the interval between surveys Or you can use a program by Oz Garton called S TOCHMVP I S TOCHMVP will calculate the probability of persistence for a metapopulation consisting of a speci ed number of Identical populations too I Or you can simulate the population using a program such as RAMAS Metapopwexe Will these predictions be 40 b or r up I 39 I Assumptions I Longterm trend continues constant u I Variance around that trend is normally distributed I No densitydependence Minimum Viable Pu ur39atrun I De nition minimum viable population for any given species in any given habitat is the smallest isolated population having a 99 chance of remaining extant for 1000 years despite the forseeable effects of demographic environmental and genetic stochasticity and natural catastrophes Shaffer 1981 Genetic Stochasticity I Deleten39ous genetic effects of very small numbers of breeders I Inbreeding depression founder effect I Loss of heterogeneity leads to accumulation of lethal genes amp effects I Ne Effective population size used to calculate this Demographic I Random effects of small numbers of individuals in the population I Example 10 anImals With 50 survival I On average 5 will be alive next ear I but with only 10 individuals its like ipping a coin Could be 6 or 4 or 7 or 3 or even 10 or 0 next year Environmental Stochasticity Natural Catastrophes I Random changes in birth and death rates produce random uctuations in rt or kt I I I Random uctuations In physrcal factors such as weather particularly temperatures and dl i quotn clln vquot f 8 389 and d due to parasites predators disease I Extreme cases of environmental stochasticity excessively high mortality I Examples I Puerto Rican Parrot populations were decimated by Hurrican Hugo a decade ago I Blackfooted Ferret almost wiped out by parvovirus Estimating MVP Persistence and MVP I Dennis et al 1991 showed that of reaching extinction threshold N 6 within time tis equal to probability of reaching threshold times conditional probability of reaching it within time t I Prob NltNe ProbTlttNltNe I p cdf umulative istn39bution 39unction of cond time to extinction I Probability of persistence to time t I 1 p cdf I Calculate the Minimum Viable Population size by nding the population size which produces the desired probability of persistence using STOCHMVP Results forLemhi I Using Shaffer s original de nition of a minimum viable population as one which provides a 99 chance of persistence for 1000 years and assuming persistence means staying above 30 spawners I then if the population were stabilized u 00001 I MVP 24 x1023 Results for Endangered Species I Species MVP I Yellowstone Grizzly Bear 3800 I Palila 61 x 1 14 I Laysan Finch 34 x 1018 I Whooping Crane 20 800 I Kirtland s Warbler 34150 I California Condor 29 x 1010 I Puerto Rican Parrot 14600 What s wrong I These are impossible I Species may never have been this abundant I But these species probably existed as a series of populations constituting one or more metapopulations Minimum Viable Metapopulation I Assuming completely independent populations in terms of environmental stochasticity I For populations the probability of at least one population persisting quot7030des Spring Chinook I UMW39 gf f 95 I f Jags e or 100 I MetaFl I 59 400 090 I 499000 3 Complications Independence of 5 1 l Palila There are 2 subpopulations for this species and there is no correlation between populations in these 2 areas r0035 n15 P902 l Spring Chinook on Salmon River Lemhi and South Fork are the 2 largest subpopulations and their rates of change are moderately correlated r0480 n27 P0011 Complications I Normal Distribution of X s vs I Densrty Dependence I Increased birth rate orlowered death rate at low population sizes Ricker type mode increases probability of persistence I Alee e ects the opposite of above decreases probablity of persistence I Spawners are only one ageclass I Holmes et a2001 showed 5yearrunning sum is a better index to total population size Stochastic Population Growth 9n quotnrr h39nnni emh39 Redd mr Mm Ti imii River Spawners Counted x WSW Chmuuk Spring Chinook 39 lm ll I z dep b r less stringent goal of 95 probability of persistence for I We of Populations I MVMetal 1330 I 2 379 I 3 197 I 5 105 Demography Demography I Future population is a result of the interplay between internal characteristics and processes and extrinsic forces and processes I We will begin by focusing on the internal factors the demographics Future Population Size Population Change I Predicting future population size requires understanding the internal workings of the population I We learn about these workings by observing changes in populations and looking for principles which would allow us to predict them I Speed of population change is referred to as r I Change results from interplay between fecundity rate mortality rate and composition age distribution and sex ratio Population Change Instantaneous Rate of I Future population size can be predicted from current population size and its rate of change I NH1 N X rate of change I NHI Nt k I k nite rate of increase I k N Nt 1 I If population is constant k 10 Change I Sometimes its better to express the nite rate of increase as an instanteous or exponential rate of increase r I k N N e 1 Ilnklogek r I If population is constant then r 0 Future Population Size Examples I If rate of change is constant then I N N0 e IlnNtlnN0rt I Equation for a straight line I y a bx Estimating rate of Rate of change quot1w cant I Finite rate of increase I k Nt1 Nt I Instantaneous rate of increase I Regress In N on t Ir In Ntln N0t I What really determines the rate of change increase or decrease in a population I Births I Deaths I Immigrants I Emigrants Rate of change Deaths Mortality I Change results from balance between rates of increase and I vecegeaasseg Births Immigrants I Decrease Deaths Emigrants IANBlDE I Immigrants and Emigrants are samll in number and depend on other populations so we ll simplify it by ignoring them for a while I What do mortality rates depend on I Age Population size Resources I Summarize mortality in a life table Life table Cohort Life Table I X age years I fX nX Survival frequency I IY Survivorship I dX Mortality I qX Mortality rate I Surw39val 0 male Song Sparrows hatched in 1976 on Mandarte Island BC Age fX l 0 115 l 1 25 l 2 19 I 3 12 I 4 2 I 5 1 I 6 0 Static Life Table Catch Constructing a Life U VU l Ages of male reindeer South Georgia Island IAge fY I0 78 I1 40 I2 19 I3 14 I4 10 l l l ldUlU talo1I l Male reindeer South Georgia Island I Age fX fx f0 1X dX I 0 78 7878 1000 0487 I1 40 4078 0513 0269 2 19 1978 0244 0064 3 14 1478 0179 0051 4 10 1078 0128 0641 I 5 5 578 0064 0064 Constructing a Life l l l l GU16 l Male reindeer South Georgia Island I Age x qX I0 8 1000 0487 0487 I 1 40 0513 0269 0525 I 2 19 0 244 0 064 0263 3 14 02179 02051 91286 10 0128 0641 I 500 5 0064 0064 1000 Graphing a Life Table Survivorship sumsva Graphing a Life Table Mortality Rate MunaW Rate Examples of Expanded Life Table Females Add fecundity mfema39 39 39 agex lAgex 1X mX l0 1000 00 00 00 I 1 08 06 048 048 l2 07 10 07 14 I3 06 10 06 18 I4 00 0 0 0 1m lmX XX XX Population I l 4 439 Ulldldblel to quotno I R Net Reproductive Rate I Sum 1 78 368 I tness Fitness l Fitness defined as 2 IX mx for a genotype I This fundamental formula is at the heart of both demographic and genetic analyses for a population I It is very clever because it combines both survivelX and fecunditymx lgitness I No daughters generation t1 I No females generation t I 2 IX mX 8 I mm 178 I I0 10 Population I G Mean length of a generation I Mean period elapsing between I birth of parents and birth of offspring I 2 IX mX X 368 I 207 I 2 IX mx 178 Population A l 1 39 A 39 Ulla39dble39 IO Elba In R In 1 78 I r 028 l G 207 Ik e equot28 132 I 1 Z e39rxlymy I Note First 2 equations are approximations but solving this last equation iteratively is exactly correct SO What I What is value of this I k and r are good descriptions for either a simple population or a more complex population I Our simple models to predict or analyze population growth might work surprisingly well IN Nt k 1 Finite Growth Rate Ik 132 I t N I0 10 I1 13 I2 17 I3 23 4 30 I5 40 Finite Growth Rate Continuous Growth I N N0 e IlnNt lnN0 rt I Plot of In NL vs tis a straightine I slope of line is r Ti Continuous Growth El 7 Is this correct I What assumptions must be true I Rate of increase k orr is constant I Stable age distribution in the population Stable Age Distribution I Lotka 1922 showed that a population that is subject to a constant schedule of birth and death rates will gradually approach a xed or stable age distribution whatever the initial age distribution may have been and will maintain this age distribution in de nitely Stable Age Distribution I CX proportion of population in age category X to X1 I Mertz 1970 showed that I CXWIX 2WX I Even if a population starts outwith a different age distribution it will attain this stable age distribution after a while Stable Age Distribution l Agex 1X k39x k39xlx CX l0 1000 10 10 044 l 1 08 0758 0606 027 I 2 07 0574 0402 018 3 06 0435 0261 011 I 4 00 0329 0 0 I Sum 2269 Is the rate of increase constant l rr resources are unrmrled sucn as rnlroduclron of Speces in new area OR rfpopualron rernsrns s2 rsrry srrnrsrnurnners over lrme r e r0 or 7 0 approxmaley nur even nere random envrronmenla vsrrsrron Wr produce random cnsnges m r or I Typrcaly resources are hmrled and rnore ammaS consurne more resources reducmg srnounr avarabe foreach anrma A Srmpe case Woud ne rf rne rare of rncresse oecrneo erh escn anrma m rne popualron Ik orr MaxkorrbNt Rate of increase declines with Nt l Ecological Models To manage fish and wildlife based on science we need to predict to predict we need to model 1 What is a model An abstraction that symbolizes the operation of processes in nature It is no more than a hypothesis clearly stated In this class we will be dealing with Stochastic models Stochastic versus deterministic model 7 There is randomness Frequentist 7 data are random events from some assumed model ie probability dist Talk about random events length of trout time series a Modeling Terms Capital letter means it s a Model Variables random Variable Response variable Y what we are interested in predicting Lowercase means it s an observation Observed data y1 y2 323 myquot Predictor variable 1 1 any auxiliary information we could use to predict Y Observed data x1 x2 x3 xquot Model Structure Probability function Function equation that maps the probability that a random draw event from a discrete random variable Y e g number of offspring number of dots on a die will take the value of y y PrY y Probability density function pdf For a continuous random variable Y length mass population density the area under the pdf equation between 2 points yL ins the probability that a random draw event y will lie between yL and yU MParameter s 1 Speci es the mathematical relationships within the model including those between the predictor variables and the response MExamples of Models Y N normal u 0392 Write on 3232 L402 plalfiameters f y normal u 0392 2gtlte 2quot variables 550 l 8 2718 pi 314 045 04 035 g 03 g 025 g 02 g 015 01 005 0 2 4 e a y What is the probability of getting a value between 6 and 7 2 Parameter Estimatation Maximum Likelihood R A Fisher 1922 We can use our observed data to find the most likely parameter values for a particular model What are we doing Trying to find the values of the parameters that maximize the probability of observing what we observed The value of the parameters that makes the likelihood as large as possible 3 Model Selection To get good predictions remember this is our goal we need a model that will closely approximate reality How can we measure how close our model is to reality ie truth Metric for measuring distance Absolute Difference AD Jy gyfy Integral means SUM Integrated squared error ISE L gy fy2 Kullback Leibler distance Kullback and Leibler 1951 KL Lgyxlngy 1nfy MNote I use gy to represent truth and y to represent an approximating model Estimates of KL Distance without knowledge of whole truth gy use observed data palticular values of Y on gy 111 Model likelihood similar to R2 Just as we used the loglikelihood to find the parameter values that got us closest to the data we can use logi value for each model to assess how close different models are to the data logi NR2 doesn t always give us a good idea of how close we are to truth because of the problem of overfitting 3 y4319x5 4o7o4x 12136x3 84117x2 76193x12a73 2 0 861 y3 9302X04736 2 o The linear regression is too biased so predictions will be poor it is underfit o underfit there are enough data to specify more meaningful relationships ie parameters 0 The fifth order polynomial IS unbiased and is in fact the true model that generated the data so why is it bad 0 Can t trust the parameter estimates 7 they re too variable to make good predictions 0 The model is overfit 7 there are not enough data to specify the relationships ie parameters 3 y22 007x274 8728gtlt103M 2 0 03 04 05 The 2nd order polynomial IS biased but it predicts better because of lower variance Bias2 Variance Number of Parameters in Model BOTH underfit and overfit models predict p00rly Because of the problem of overfitting we cannot use the logi r2 to measure how close we are to the truth 112 Information theoretic criteria ITC Idea is that we CAN use the log i to estimate the relative KL distance IF we correct for over tting 1121 Akaike s Information Criterion AIC AIC 2log 21lt
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