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Published byJulianna Veronica Newman Modified about 1 year ago

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An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model

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Generic Model f(x) prey growth term g(y) predator mortality term h(x,y) predation term e prey into predator biomass conversion coefficient

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Lotka-Volterra Model r prey growth rate : Malthus law m predator mortality rate : natural mortality Mass action law a and b predation coefficients : b=ea e prey into predator biomass conversion coefficient

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Lotka-Volterra nullclines

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Direction field for Lotka-Volterra model

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Local stability analysis Jacobian at positive equilibrium detJ*>0 and trJ*=0 (center)

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Linear 2D systems (hyperbolic)

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Local stability analysis Proof of existence of center trajectories (linearization theorem) Existence of a first integral H(x,y) :

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Lotka-Volterra model

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Hare-Lynx data (Canada)

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Logistic growth (sheep in Australia)

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Lotka-Volterra Model with prey logistic growth

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Nullclines for the Lotka-Volterra model with prey logistic growth

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Lotka-Volterra Model with prey logistic growth Equilibrium points : (0,0) (K,0) (x*,y*)

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Local stability analysis Jacobian at positive equilibrium detJ*>0 and trJ*<0 (stable)

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Condition for local asymptotic stability

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Lotka-Volterra model with prey logistic growth : coexistence

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Lotka-Volterra with prey logistic growth : predator extinction

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Transcritical bifurcation (K,0) stable and (x*,y*) unstable and negative (K,0) and (x*,y*) same (K,0) unstable and (x*,y*) stable and positive

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Loss of periodic solutions coexistencePredator extinction

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Functional response I and II

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Holling Model

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Existence of limit cycle (Supercritical Hopf bifurcation) Polar coordinates

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Stable equilibrium

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At bifurcation

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Existence of a limit cycle

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Supercritical Hopf bifurcation

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Poincaré-Bendixson Theorem A bounded semi-orbit in the plane tends to : a stable equilibrium a limit cycle a cycle graph

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Trapping region

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Trapping region : Annulus

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Example of a trapping region Van der Pol model ( >0)

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Holling Model

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Nullclines for Holling model

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Poincaré box for Holling model

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Holling model with limit cycle

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Paradox of enrichment When K increases : Predator extinction Prey-predator coexistence (TC) Prey-predator equilibrium becomes unstable (Hopf) Occurrence of a stable limit cycle (large variations)

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Other prey-predator models Functional responses (Type III, ratio-dependent …) Prey-predator-super-predator… Trophic levels

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Routh-Hurwitz stability conditions Characteristic equations Stability conditions : M* l.a.s.

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Routh-Hurwitz stability conditions Dimension 2 Dimension 3

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3-trophic example

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Interspecific competition Model Transformed system

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Competition model

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