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title. Trapper Drives Lynx to Eat Hare. Bo Deng February 2006 UNL. title. 2d Model. Comprehensiv Predator-Prey Model. H --- Hare Population L --- Lynx Population r = b 1 – d 1 --- Hare Per-Capita Intrinsic Growth Rate

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  1. title Trapper Drives Lynx to Eat Hare Bo Deng February 2006 UNL

  2. title • 2d Model Comprehensiv Predator-Prey Model H ---Hare Population L ---Lynx Population r = b1 – d1---Hare Per-Capita Intrinsic Growth Rate K =m1/ b1 – d1---Hare Carrying Capacity mi---Interspecific Competition Parameters bi---Birth-to-Consumption Ratio di---Natural Per-Capita Death Rate ai---Probability Rate of Discovery hi---Handling Time Per-Prey

  3. Basic Models • 2d Model

  4. Elton & Nichoslon 1942 (Odum 1953) Hare Eats Lynx Cycle Guilpin’s Disease Explanation 1973

  5. title • 3d Model HLT Model 1 H ---Hare Population L ---Lynx PopulationT ---Trapper Population b3---Fur Collection-to-Recruitment Ratio d3---Per-Capita Retirement Rate m3---Bankruptcy or Consolidation Rate h2---Handling Time Per-Catch

  6. title • 3d Model Assumption: Hare capacity is mediated by trapper K = K (T ) =K0 (T + T0) / (T + T1) K K0 0 T HLT Model 2 H ---Hare Population L ---Lynx PopulationT ---Trapper Population K(T)---Trapper Mediated Capacity b3---Fur Collection-to-Recruitment Ratio d3---Per-Capita Retirement Rate m3---Bankruptcy Rate h3---Handling Time Per-Catch

  7. Basic Models • Comparison K K K0 K0 0 T 0 T H-L Cycle L-H Cycle Hare and Trapper oscillate in-phase Hare and Trapper oscillate out-phase

  8. W.M. Shaffer `… Evidence for a Strange Attractor in Nature?’ Amer. Nat. 1984 Time-Delayed Embedding Method: F. Takens, 1981

  9. Field Studies Lyapunov Exponent Logistic Map Def. of Lyapunov Exponent: Working Def of Chaos: Attractor + Positive LE = Chaos

  10. Field Data S. Ellner & P. Turchin Amer. Nat. 1995 Lyapunov Number > 1  Chaos

  11. Field Data S. Ellner & P. Turchin Amer. Nat. 1995 Lyapunov Number > 1  Chaos

  12. Lab Data S. Ellner & P. Turchin Amer. Nat. 1995

  13. Trends in Eco. & Evol. 1989 Is ecological chaos an evolutionary cruel and inhuman punishment?

  14. title • Equilibrium Principles Equilibrium Principles L L’=0 H ---Hare Population L ---Lynx PopulationT ---Trapper Population b3---Fur Collection-to-Recruitment Ratio d3---Per-Capita Retirement Rate m3---Bankruptcy Rate h2---Handling Time Per-Catch H ’=0 0 H (b1 - d1 )/ m1 Theorem:(Enrichment Equilibrium Principle) For sufficiently large b1 / h0 – d1 , the equilibrium point with the largest top-predator density and the largest predator density is always stable.

  15. Equilibrium Principles

  16. title • Equilibrium Principles L L’=0 H ’=0 H ---Hare Population L ---Lynx PopulationT ---Trapper Population b3---Fur Collection-to-Recruitment Ratio d3---Per-Capita Retirement Rate m3---Bankruptcy Rate h2---Handling Time Per-Catch 0 H Theorem:(Efficiency Equilibrium Principle) For sufficiently large b3 / h2 – d3, the equilibrium point with the largest, positive top-predator density is always stable. If there is no equilibrium points of positive top-predator density, then the equilibrium point with the largest predator density is stable for sufficiently large b2 / h1 – d2

  17. Equilibrium Principles

  18. Other Examples -- Childhood Epidemics SEIR (Susceptible, Exposed, Infectious, Recovered) Dynamics W.M. Schaffer, L.F. Olsen, G.L. Truty, S.L. Fulmer 1985, 1993

  19. ... More Childhood Epidemics W.M. Schaffer, L.F. Olsen, G.L. Truty, S.L. Fulmer 1985, 1993

  20. Canadian Snowshoe Hare -Lynx Dynamics 1844 -- 1935 N.C. Stenseth Science 1995

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