AIMS Prey predator Models

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


Slide2 : Generic Model f(x) prey growth term g(y) predator mortality term h(x,y) predation term e prey into predator biomass conversion coefficient


Slide3 : 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


Slide4 : Lotka-Volterra nullclines


Direction field for Lotka-Volterra model : Direction field for Lotka-Volterra model


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


Slide7 : Linear 2D systems (hyperbolic)


Local stability analysis : 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


Lotka-Volterra model : Lotka-Volterra model


Hare-Lynx data (Canada) : Hare-Lynx data (Canada)


Slide12 : Logistic growth (sheep in Australia)


Slide13 : Lotka-Volterra Model with prey logistic growth


Nullclines for the Lotka-Volterra model with prey logistic growth : Nullclines for the Lotka-Volterra model with prey logistic growth


Slide15 : Lotka-Volterra Model with prey logistic growth Equilibrium points : (0,0) (K,0) (x*,y*)


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


Slide17 : Condition for local asymptotic stability


Lotka-Volterra model with prey logistic growth : coexistence : Lotka-Volterra model with prey logistic growth : coexistence


Lotka-Volterra with prey logistic growth : predator extinction : Lotka-Volterra with prey logistic growth : predator extinction


Slide20 : 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


Slide21 : Loss of periodic solutions coexistence Predator extinction


Functional response I and II : Functional response I and II


Slide23 : Holling Model


Slide24 : Existence of limit cycle (Supercritical Hopf bifurcation) Polar coordinates


Slide25 : Stable equilibrium


Slide26 : At bifurcation


Slide27 : Existence of a limit cycle


Slide28 : Supercritical Hopf bifurcation


Poincaré-Bendixson Theorem : 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


Trapping region : Annulus : Trapping region : Annulus


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


Slide33 : Holling Model


Nullclines for Holling model : Nullclines for Holling model


Poincaré box for Holling model : Poincaré box for Holling model


Holling model with limit cycle : Holling model with limit cycle


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


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


Routh-Hurwitz stability conditions : Routh-Hurwitz stability conditions Dimension 2 Dimension 3


3-trophic example : 3-trophic example


Slide42 : Interspecific competition Model Transformed system


Competition model : Competition model