Bifurcations
A bifurcation occurs when a gradual change in a parameter causes a qualitative change in the behaviour of a dynamical system.
Parameter-dependent system
Consider
\[\frac{dx}{dt}=f(x;\mu),\]where \(\mu\) is a parameter. As \(\mu\) changes, equilibria can appear, disappear, exchange stability or give rise to oscillations.
Common types
Important examples include saddle-node, transcritical, pitchfork and Hopf bifurcations. Each represents a different way in which the structure of the dynamics changes.
Biological importance
Bifurcations can represent thresholds such as disease invasion, collapse of a population, emergence of oscillations or switching between alternative stable states.
Key idea. A bifurcation marks a parameter value at which the qualitative long-term behaviour of the model changes.