Bifurcations
A bifurcation occurs when changing a parameter causes a qualitative change in the behaviour of a dynamical system. The parameter may change gradually, but at a critical value the structure of the long-term dynamics can change.
Parameters versus state variables
Consider
\[\frac{dx}{dt}=f(x;\mu).\]The state variable \(x(t)\) changes with time. The quantity \(\mu\) is a parameter. Bifurcation analysis asks how equilibria and their stability change as \(\mu\) varies.
What is a bifurcation diagram?
The horizontal axis shows the parameter and the vertical axis shows an equilibrium value or, for oscillations, an amplitude. In the diagrams below, solid curves denote stable branches and dashed curves denote unstable branches.
1. Saddle-node bifurcation
Consider
\[\frac{dx}{dt}=\mu-x^2.\]Equilibria satisfy \(x^*=\pm\sqrt{\mu}\). For \(\mu<0\), there are no real equilibria. At \(\mu=0\), the two branches meet. For \(\mu>0\), one stable and one unstable equilibrium exist. Since \(f'(x)=-2x\), the upper branch is stable and the lower branch unstable.
2. Transcritical bifurcation
For
\[\frac{dx}{dt}=\mu x-x^2=x(\mu-x),\]the equilibria are \(x^*=0\) and \(x^*=\mu\). They cross at \(\mu=0\) and exchange stability.
Disease-free and endemic equilibria often exchange stability at an invasion threshold. In many epidemic models this is associated with \(R_0=1\), although the exact bifurcation structure depends on the model.
3. Pitchfork bifurcation
A standard supercritical pitchfork is
\[\frac{dx}{dt}=\mu x-x^3.\]The equilibrium \(x^*=0\) exists for all \(\mu\). When \(\mu>0\), two additional equilibria \(x^*=\pm\sqrt{\mu}\) appear. The central branch loses stability while the two new branches are stable.
Exact pitchfork bifurcations are closely associated with symmetry, so biological asymmetry can alter this idealised picture.
4. Hopf bifurcation
A Hopf bifurcation occurs in a system of at least two dimensions when a complex-conjugate pair of eigenvalues crosses the imaginary axis:
\[\lambda_{1,2}=a(\mu)\pm ib(\mu).\]At the critical value the real part passes through zero. Under the additional nondegeneracy conditions for a Hopf bifurcation, periodic solutions arise near the equilibrium. In a supercritical Hopf bifurcation, a stable equilibrium typically loses stability while a small stable periodic orbit emerges.
Why Hopf bifurcations matter in biology
They can mark the onset of sustained oscillations in predator–prey systems, host–parasite models, biochemical networks and epidemic models.
Four common bifurcations compared
| Type | Main change | Biological interpretation |
|---|---|---|
| Saddle-node | stable and unstable equilibria collide | tipping point or loss of a viable state |
| Transcritical | two equilibria exchange stability | invasion threshold |
| Pitchfork | one branch changes stability and two branches emerge | symmetry breaking |
| Hopf | equilibrium changes stability and nearby periodic solutions arise | onset or loss of sustained oscillation |
Connection with eigenvalues
Bifurcations are often detected by following Jacobian eigenvalues as a parameter changes. For common saddle-node, transcritical and pitchfork bifurcations, a real eigenvalue passes through zero. For a Hopf bifurcation, a complex pair crosses the imaginary axis.
Critical slowing down
As a dominant stable eigenvalue approaches zero from below, recovery from a disturbance becomes slower:
\[\lambda\to0^-\quad\Rightarrow\quad e^{\lambda t}\text{ decays more slowly}.\]This is called critical slowing down. It can occur near some transitions but is not a universal early-warning signal.
Bifurcation versus threshold
Not every threshold is a bifurcation. A mathematical bifurcation requires a qualitative change in the structure or stability of the dynamics.
Bifurcation versus stochastic switching
A deterministic bifurcation occurs because a parameter changes. In a stochastic system, random fluctuations can sometimes move a system between states while parameters remain fixed. These are different mechanisms.
A practical workflow
Choose a biologically meaningful parameter. Find equilibria as functions of that parameter. Evaluate the Jacobian and determine stability. Plot equilibrium branches, using a clearly stated line convention for stable and unstable branches. Identify where branches meet or stability changes, then interpret those critical values biologically.