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