16.6 Transcritical Bifurcations: When States Exchange Their Stability

The saddle-node bifurcation introduced in the previous section describes the appearance or disappearance of equilibrium points. Not all biological transitions, however, occur in this way.

In some systems, the equilibrium points themselves remain present, but their biological roles change dramatically. A state that was previously stable may become unstable, while another equilibrium simultaneously becomes stable.

Rather than appearing or disappearing, the two equilibria exchange their stability.

This phenomenon is known as a transcritical bifurcation.

Competition between biological states

Transcritical bifurcations often arise when two competing biological states coexist.

For example, consider the spread of an infectious disease.

One equilibrium corresponds to a disease-free population.

The second equilibrium represents the persistence of the pathogen.

When the pathogen is unable to spread efficiently, the disease-free equilibrium is stable, whereas the endemic equilibrium is unstable.

As the transmission rate increases, both equilibria remain present, but their stability changes.

Beyond a critical threshold, the endemic equilibrium becomes stable and the disease-free equilibrium becomes unstable.

The system therefore undergoes a qualitative transition without creating or destroying equilibrium points.

Biological interpretation

Unlike the saddle-node bifurcation, where equilibria disappear completely, the transcritical bifurcation represents a competition between alternative biological states.

The system always possesses both equilibria.

What changes is which equilibrium attracts nearby trajectories.

This mechanism is observed in many biological contexts, including

host-pathogen interactions,
species competition,
invasion of ecological communities,
regulatory systems with competing states.

In each case, changing a biological parameter alters which state dominates the long-term behaviour of the system.

Key concepts

In a transcritical bifurcation, equilibrium points remain present.
Stable and unstable equilibria exchange their stability.
Transcritical bifurcations commonly arise in systems involving biological competition.
The qualitative behaviour changes even though the number of equilibria remains constant.

Summary

Transcritical bifurcations describe situations in which two existing equilibrium points exchange their stability as a parameter changes. Rather than creating or destroying equilibria, this type of bifurcation determines which biological state becomes dominant under different environmental or physiological conditions.

Self-check questions

How does a transcritical bifurcation differ from a saddle-node bifurcation?
What changes during a transcritical bifurcation?
Why are transcritical bifurcations common in competitive biological systems?
Give two biological examples of transcritical bifurcations.
Why does the number of equilibrium points remain unchanged?