Bifurcations of 2D Systems: Understanding Dynamics in Ecological Models

1. Mini-course bifurcation theory Part three: bifurcations of 2D systems George van Voorn

2. Bifurcations • Bifurcations of equilibria in 1D • Transcritical (λ = 0) • Tangent (λ = 0) • In 2D • Transcritical (One λ = 0) • Tangent (One λ = 0)

3. Example • 2D ODE Rosenzweig-MacArthur (1963) R = intrinsic growth rate K = carrying capacity A/B = searching and handling C = yield D = death rate

4. Example • At K = 0  x = 0, y = 0 • 0 < K < 6  x > 0, y = 0 • At K = 6 x = K, y = 0 • 6 < K < ?  x = K, y > 0 • K = 0 and K = 6  transcritical • Invasion criterion species x,y

5. Hopf bifurcation • Andronov-Hopf bifurcation: • Transition from stable spiral to unstable spiral (equilibrium becomes unstable) • Or vice versa • Periodic orbit (limit cycle), stable • Or vice verse, respectively • Conditions:

6. x,y α Hopf bifurcation

7. Example • At K≈ 20  x = K, y > 0 • Limit cycle • Change in dynamics of species x,y

8. Paradox of enrichment • Isoclines RM model x = K, for all K > 6  increase only in y

9. Paradox of enrichment One-parameter bifurcation diagram

10. Paradox of enrichment Maximal values x and y Minimal values x and y Equilibria One-parameter bifurcation diagram

11. Paradox of enrichment Oscillations increase in size for larger K

12. Paradox of enrichment • Large oscillations  extinction probabilities • Paradox: • Increase in food availability K • No benefit for prey x • Increased probability of extinction system

13. Limit cycle bifurcations • For limit cycles same bifurcations as for equilibria • Imagine cross section

14. Limit cycle bifurcations • Transcritical  Cycle on axis (mostly) • Tangent  Birth or destruction cycle(s)

15. Limit cycle bifurcations • Hopf (called Neimark-Sacker)  Torus

16. Limit cycle bifurcations • Flip bifurcation • Manifold around cycle

17. Flip bifurcations • Manifold twisted

18. Flip bifurcations • Flip bifurcation of limit cycle • Manifold twisted (Möbius ribbon) • Period doubling

19. Codim 2 points • Bifurcation points can be continued in two-parameter space = bifurcation curve • Continuation can result in: • Bifurcation points of higher co-dimension

20. Codim 2 points • Bogdanov-Takens • Cusp • Generalised Hopf (Bautin)

21. Example • Bazykin model • Calculate equilibrium • Vary one parameter until a bifurcation is encountered

22. Bazykin x*,y* Continuation in two-parameter space

23. Bazykin: dynamics Stable node: coexistence No positive equilibria: extinction Stable cycle: coexistence Unstable equilibria: extinction

24. Bazykin: BT point Bogdanov-Takens point  tangent & Hopf

25. Bazykin: GH point Bautin point  transition Hopf from stable to unstable point

26. Bazykin: cusp point Cusp point  collision two tangent points

27. Question What happens here? Stable cycle: coexistence Unstable equilibria: extinction

28. Global bifurcations • BT point: origin of homoclinic bifurcation

29. Bazykin: homoclinic Starting at Hopf continue cycle. What happens?

30. Bazykin: homoclinic Limit cycle period to infinity. Why?

31. Homoclinic connection Wu Ws Homoclinic connecting orbit: Wu = Ws Time to infinity near equilibrium

32. Heteroclinic connection Ŵs Wu Heteroclinic connecting orbit: Wu = Ŵs

33. Bazykin: homoclinic