AIMS Prey predator Models

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Published on January 1, 2008

Author: Freedom

Source: authorstream.com

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

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