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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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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**

coexistence Predator 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 (l>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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