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Mini-course bifurcation theory
George van Voorn
Part one: introduction, 1D systems
Introduction
• One-dimensional systems– Notation & Equilibria– Bifurcations
• Two-dimensional systems– Equilibria– Eigenfunctions– Isoclines & manifolds
Introduction
• Two-dimensional systems– Bifurcations of equilibria– Limit cycles– Bifurcations of limit cycles– Bifurcations of higher co-dimension– Global bifurcations
Introduction
• Multi-dimensional systems– Example: Rosenzweig-MacArthur (3D)– Equilibria/stability– Local bifurcation diagram– Chaos– Boundaries of chaos
Introduction
• Goal– Very limited amount of mathematics– Biological interpretation of bifurcations– Questions?!
Systems & equilibria
• One-dimensional ODE
• Autonomous (time dependent)• Equilibria: equation equals zero
Stability
• Equilibrium stability– Derivative at equilibrium
– Stable
– Unstable
Bifurcation
• Consider a parameter dependent system
• If change in parameter– Structurally stable: no significant change– Bifurcation: sudden change in dynamics
Transcritical
• Consider the ODE
• Two equilibria
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)
Transcritical
Transcritical bifurcation point α = 0
Tangent
• Consider the ODE
• Two equilibria (α > 0)
Tangent
Tangent bifurcation point α = 0
Application
• Model by Rietkerk et al., Oikos 80, 1997• Herbivory on vegetation in semi-arid regions
P = plantsg(N) = growth functionb = amount of herbivoryd = mortality
ApplicationSay, the model bears realism, then possible measurement points
ApplicationWould this have been a Nature article …
Application
TC
T
But:
Application
TC
T
bistability extinctieequilibrium
Application
1
2
3
4
1. Man wants more2. Sudden extinction3. Significant decrease in exploitation necessary4. Recovery
Recovery from an ecological (anthropogenic) disaster:
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