An introduction to prey-predator Models

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. Lotka-Volterra model

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 coexistence Predator 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 (l>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