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UI WLF 448 - Lotka‐Volterra Model

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Lotka‐Volterra ModelLotkaVolterra ModelH = number of preydH/dt=rH‐bHPpyr = prey population growth rateb = attack ratedH/dt r H bHP P = number of predatorsdP/dt =cHPkPc = predator population growth rate due to predationk = rate of predator decline in dP/dt = cHP‐k P pabsence of preyLotka‐Volterra ModelLotkaVolterra ModeldH / dt < 0dP / dt < 0dH / dt < 0dP / dt > 0Pr/bPk/cdH / dt > 0dP / dt > 0dH / dt > 0dP / dt < 0HModified Lotka‐Volterra ModelModified LotkaVolterra ModeldH / dt = r H (1‐H/K) – aHP / (1+aHh)dP/dt = cP (1‐(PJ/H)) –k PH = number of preyr = prey population growth rateb = attack rateP = number of predatorsc=predator population growth rate due to predationc predator population growth rate due to predationk = rate of predator decline in absence of preyJ = prey density required to support 1 predator per areaStabilityStability• Tanner (1975 Ecology 56:855)• Explored features of this model to find general properties, particularly model stability• Does the “critical point” where predator and prey isoclines cross produce a:py p– stable equilibrium (“focus point”)–limit cyclelimit cycle– unstable •predator growth / prey growth rates (s/r)•predator growth / prey growth rates (s/r) (note c = s)Tanner (1975)Tanner (1975)Stable focus when the critical point falls to the right offalls to the right of the prey zero isocline peak for all values of s/rTanner (1975)Tanner (1975)When the critical point falls to the left of the prey zeroleft of the prey zero isocline peak,1) stable focus if s/r is largeTanner (1975)Tanner (1975)When the critical point falls to the left of the prey zeroleft of the prey zero isocline peak, 2) limit cycle if s/r smallTanner (1975)Tanner (1975)When the critical point falls to the left of the prey zeroleft of the prey zero isocline peak, 3) unstable focus if s/r small and K is very large –ygextinction; no coexistenceTanner 1975Tanner 1975•What if predator is limited a resource that isWhat if predator is limited a resource that is independent of both predators and prey such as nest sites or space rather than prey oras nest sites or space rather than prey or predator numbers?P PH HTanner (1975)Tanner (1975)Once again, since the critical point falls to the right offalls to the right of the prey zero isocline peak, a stable focus results for all values of s/rTanner (1975)Tanner (1975)Again, since the critical point falls to the right of thethe right of the prey zero isocline peak, a stable results for all values of s/rTanner (1975)Tanner (1975)When the predator and prey zero Unstable focusisoclines cross three time, two stable and one Stable focusstable and oneunstable (“saddle”) points are created. Population canPopulation can “jump” from one to the other Stable focusdepending on starting point and other model constants orTanner (1975)Tanner (1975)The prey population can get Unstable focus“stuck” at very low density unless predation rates Stable focuspredation ratesdrop substantially , called a predator pitpitStable focus“P d t Pit”“Predator Pit”Tanner 1975Tanner 1975•Complex model behavior nearly any outcome!Complex model behavior, nearly any outcome!• So what? Is this useful?Tanner 1975Tanner 1975•Complex model behavior nearly any outcome!Complex model behavior, nearly any outcome!• So what? Is this useful?fl d h l f• Tanner reflected on the general patterns from models• Hypothesized that stable prey species were either strongly self‐limited (e.g., by territoriality) or the prey population growth rate was less than that of the predator• How would you test?Tanner 1975Tanner 1975•Hypothesized that stable prey species wereHypothesized that stable prey species were either strongly self‐limited (e.g., by territoriality) or the prey population growthterritoriality), or the prey population growth rate was less than that of the predator–Prey growth rate appeared higher (s/r < 1) for:–Prey growth rate appeared higher (s/r < 1) for:• sparrow hawk / house sparrow and •Mink / muskratMink / muskrat– And both prey species thought to be self‐limited (sparrows: food or breeding sites; muskrats: (p g ;territories)Tanner 1975Tanner 1975•Hypothesized that stable prey species wereHypothesized that stable prey species were either strongly self‐limited (e.g., by territoriality) or the prey population growthterritoriality), or the prey population growth rate was less than that of the predator–Prey growth rate appeared similar(s/r = 1) for–Prey growth rate appeared similar(s/r = 1) for• Lynx / snowshoe hare–Hare and lynx show cyclesHare and lynx show cyclesTanner 1975Tanner 1975•Hypothesized that stable prey species wereHypothesized that stable prey species were either strongly self‐limited (e.g., by territoriality) or the prey population growthterritoriality), or the prey population growth rate was less than that of the predator–Prey growth rate appeared lower(s/r > 1) for–Prey growth rate appeared lower(s/r > 1) for several prey species with weak self‐regulation:• Mt. lion / mule deerto/uedee• Wolf / (moose, caribou, WT deer, white sheep)Model assumptionsModel assumptions• No time lags• No prey refuges•Predator searching constant, not affected byPredator searching constant, not affected by external factors•No differences in prey susceptibility•No differences in prey susceptibilityOptimal Foraging TheoryOptimal Foraging Theory•How does a predator choose which prey to hunt forHow does a predator choose which prey to hunt for and for how long?• Theory developed to identify the optimal choices yp ypbased 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 profitabilitieschange through timeModel assumptionsModel assumptions• No time lags• No prey refuges•Predator searching constant, not affected byPredator searching constant, not affected by external factors•No differences in prey susceptibility•No differences in prey susceptibility• Prey switching and switching of habitats–Predators switching to another prey at low prey density essentially creates a refuge, theoretically increasing stabilityincreasing stability.– Evidence?Hanski et al. 1993 and Turchin and Hanski•Vole population dynamics in northern EuropeVole


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UI WLF 448 - Lotka‐Volterra Model

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