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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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Presentation on theme: "An introduction to prey-predator Models Lotka-Volterra model Lotka-Volterra model with prey logistic growth Holling type II model."— Presentation transcript:

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

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

3 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

4 Lotka-Volterra nullclines

5 Direction field for Lotka-Volterra model

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

7 Linear 2D systems (hyperbolic)

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

9 Lotka-Volterra model

10

11 Hare-Lynx data (Canada)

12 Logistic growth (sheep in Australia)

13 Lotka-Volterra Model with prey logistic growth

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

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

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

17 Condition for local asymptotic stability

18 Lotka-Volterra model with prey logistic growth : coexistence

19 Lotka-Volterra with prey logistic growth : predator extinction

20 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

21 Loss of periodic solutions coexistencePredator extinction

22 Functional response I and II

23 Holling Model

24 Existence of limit cycle (Supercritical Hopf bifurcation) Polar coordinates

25 Stable equilibrium

26 At bifurcation

27 Existence of a limit cycle

28 Supercritical Hopf bifurcation

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

30 Trapping region

31 Trapping region : Annulus

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

33 Holling Model

34 Nullclines for Holling model

35 Poincaré box for Holling model

36 Holling model with limit cycle

37 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)

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

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

40 Routh-Hurwitz stability conditions Dimension 2 Dimension 3

41 3-trophic example

42 Interspecific competition Model Transformed system

43 Competition model


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