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

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

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

Lotka-Volterra nullclines

Direction field for Lotka-Volterra model

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

Linear 2D systems (hyperbolic)

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

Lotka-Volterra model

Lotka-Volterra model

Hare-Lynx data (Canada)

Logistic growth (sheep in Australia)

Lotka-Volterra Model with prey logistic growth

Nullclines for the Lotka-Volterra model with prey logistic growth

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

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

Condition for local asymptotic stability

Lotka-Volterra model with prey logistic growth : coexistence

Lotka-Volterra with prey logistic growth : predator extinction

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

Loss of periodic solutions coexistence Predator extinction

Functional response I and II

Holling Model

Existence of limit cycle (Supercritical Hopf bifurcation) Polar coordinates

Stable equilibrium

At bifurcation

Existence of a limit cycle

Supercritical Hopf bifurcation

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

Trapping region

Trapping region : Annulus

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

Holling Model

Nullclines for Holling model

Poincaré box for Holling model

Holling model with limit cycle

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)

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

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

Routh-Hurwitz stability conditions Dimension 2 Dimension 3

3-trophic example

Interspecific competition Model Transformed system

Competition model