title Bo Deng February 2006 UNL

title H --- Hare Population L --- Lynx Population r = b 1 – d Hare Per-Capita Intrinsic Growth Rate K = m 1 / b 1 – d Hare Carrying Capacity m i --- Interspecific Competition Parameters b i --- Birth-to-Consumption Ratio d i --- Natural Per-Capita Death Rate a i --- Probability Rate of Discovery h i --- Handling Time Per-Prey H --- Hare Population L --- Lynx Population r = b 1 – d Hare Per-Capita Intrinsic Growth Rate K = m 1 / b 1 – d Hare Carrying Capacity m i --- Interspecific Competition Parameters b i --- Birth-to-Consumption Ratio d i --- Natural Per-Capita Death Rate a i --- Probability Rate of Discovery h i --- Handling Time Per-Prey 2d Model

Basic Models 2d Model

Guilpin’s Disease Explanation 1973 Elton & Nichoslon 1942 (Odum 1953)

title H --- Hare Population L --- Lynx Population T --- Trapper Population b Fur Collection-to-Recruitment Ratio d Per-Capita Retirement Rate m Bankruptcy or Consolidation Rate h Handling Time Per-Catch H --- Hare Population L --- Lynx Population T --- Trapper Population b Fur Collection-to-Recruitment Ratio d Per-Capita Retirement Rate m Bankruptcy or Consolidation Rate h Handling Time Per-Catch 3d Model

title H --- Hare Population L --- Lynx Population T --- Trapper Population K(T) --- Trapper Mediated Capacity b Fur Collection-to-Recruitment Ratio d Per-Capita Retirement Rate m Bankruptcy Rate h Handling Time Per-Catch H --- Hare Population L --- Lynx Population T --- Trapper Population K(T) --- Trapper Mediated Capacity b Fur Collection-to-Recruitment Ratio d Per-Capita Retirement Rate m Bankruptcy Rate h Handling Time Per-Catch 3d Model Assumption: Hare capacity is mediated by trapper K = K (T ) = K 0 (T + T 0 ) / (T + T 1 ) Assumption: Hare capacity is mediated by trapper K = K (T ) = K 0 (T + T 0 ) / (T + T 1 ) T K K0K0 0

Comparison Basic Models T K K0K0 0 Hare and Trapper oscillate in-phaseHare and Trapper oscillate out-phase T K K0K0 0

W.M. Shaffer `… Evidence for a Strange Attractor in Nature?’ Amer. Nat Method: F. Takens, 1981

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

S. Ellner & P. Turchin Amer. Nat Lyapunov Number > 1  Chaos

S. Ellner & P. Turchin Amer. Nat Lyapunov Number > 1  Chaos

S. Ellner & P. Turchin Amer. Nat. 1995

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

title H --- Hare Population L --- Lynx Population T --- Trapper Population b Fur Collection-to-Recruitment Ratio d Per-Capita Retirement Rate m Bankruptcy Rate h Handling Time Per-Catch H --- Hare Population L --- Lynx Population T --- Trapper Population b Fur Collection-to-Recruitment Ratio d Per-Capita Retirement Rate m Bankruptcy Rate h Handling Time Per-Catch Equilibrium Principles Theorem: (Enrichment Equilibrium Principle) For sufficiently large b 1 / h 0 – d 1, the equilibrium point with the largest top-predator density and the largest predator density is always stable. Theorem: (Enrichment Equilibrium Principle) For sufficiently large b 1 / h 0 – d 1, the equilibrium point with the largest top-predator density and the largest predator density is always stable. H L 0 (b 1 - d 1 )/ m 1 L’ = 0 H ’ = 0

Equilibrium Principles

title H --- Hare Population L --- Lynx Population T --- Trapper Population b Fur Collection-to-Recruitment Ratio d Per-Capita Retirement Rate m Bankruptcy Rate h Handling Time Per-Catch H --- Hare Population L --- Lynx Population T --- Trapper Population b Fur Collection-to-Recruitment Ratio d Per-Capita Retirement Rate m Bankruptcy Rate h Handling Time Per-Catch Equilibrium Principles Theorem: (Efficiency Equilibrium Principle) For sufficiently large b 3 / h 2 – d 3, 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 b 2 / h 1 – d 2 Theorem: (Efficiency Equilibrium Principle) For sufficiently large b 3 / h 2 – d 3, 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 b 2 / h 1 – d 2 H L 0 L’ = 0 H ’ = 0

Equilibrium Principles

SEIR (Susceptible, Exposed, Infectious, Recovered) Dynamics W.M. Schaffer, L.F. Olsen, G.L. Truty, S.L. Fulmer 1985, 1993

W.M. Schaffer, L.F. Olsen, G.L. Truty, S.L. Fulmer 1985, 1993

N.C. Stenseth Science