1. Mini-course bifurcation theory George van Voorn Part one: introduction, 1D systems

2. Introduction One-dimensional systems –Notation & Equilibria –Bifurcations Two-dimensional systems –Equilibria –Eigenfunctions –Isoclines & manifolds

3. Introduction Two-dimensional systems –Bifurcations of equilibria –Limit cycles –Bifurcations of limit cycles –Bifurcations of higher co-dimension –Global bifurcations

4. Introduction Multi-dimensional systems –Example: Rosenzweig-MacArthur (3D) –Equilibria/stability –Local bifurcation diagram –Chaos –Boundaries of chaos

5. Introduction Goal –Very limited amount of mathematics –Biological interpretation of bifurcations –Questions?!

6. Systems & equilibria One-dimensional ODE Autonomous (time dependent) Equilibria: equation equals zero

7. Stability Equilibrium stability –Derivative at equilibrium –Stable –Unstable

8. Bifurcation Consider a parameter dependent system If change in parameter –Structurally stable: no significant change –Bifurcation: sudden change in dynamics

9. Transcritical Consider the ODE Two equilibria

10. Transcritical Example: α = 1 Equilibria: x = 0, x = 1 Derivative: –2x + α Stability –x = 0  f ’(x) > 0 (unstable) –x = α  f ’(x) < 0 (stable)

11. Transcritical Transcritical bifurcation point α = 0

12. Tangent Consider the ODE Two equilibria (α > 0)

13. Tangent Tangent bifurcation point α = 0

14. Application Model by Rietkerk et al., Oikos 80, 1997 Herbivory on vegetation in semi-arid regions P = plants g(N) = growth function b = amount of herbivory d = mortality

15. Application Say, the model bears realism, then possible measurement points

16. Application Would this have been a Nature article …

17. Application TC T But:

18. Application TC T bistabilityextinctieequilibrium

19. Application 1 2 3 4 1. Man wants more 2. Sudden extinction 3. Significant decrease in exploitation necessary 4. Recovery Recovery from an ecological (anthropogenic) disaster:

20. Application If increase in level of herbivory (b) Extinction of plants (P) might follow Recovery however requires a much lower b Bifurcation analysis as a useful tool to analyse models