15.8 When Equilibria Change

Throughout this chapter, we have assumed that the mathematical model describing the biological system remains unchanged. Under this assumption, we identified equilibrium points, determined whether they are stable or unstable, and investigated how trajectories evolve towards or away from these states.

In reality, however, biological systems rarely operate under constant conditions.

Environmental factors such as temperature, nutrient availability, light intensity, or stress continuously change. Likewise, biological parameters—including reaction rates, protein abundances, or signalling strengths—may vary during development or in response to external stimuli. As these conditions change, the underlying dynamics of the system also change.

This raises a new and fundamental question:

What happens when the equilibrium itself changes?

Equilibria are not fixed

An equilibrium is not an intrinsic property of a biological system. Instead, it depends on the parameters that govern the system’s behaviour.

Consider a population living in an environment with abundant resources. The carrying capacity may be large, allowing the population to stabilize at a high abundance. If the environment deteriorates—for example because of habitat loss or climate change—the carrying capacity decreases. The equilibrium population size therefore shifts to a lower value.

Many biological systems behave in a similar way.

Changes in enzyme activity alter metabolic steady states. Mutations modify gene regulatory networks. Hormonal signals reshape developmental pathways. In each case, the location and stability of the equilibrium depend on the underlying biological parameters.

Small parameter changes can have large consequences

Often, gradual changes in biological parameters produce only gradual changes in system behaviour.

Sometimes, however, a very small parameter change produces a dramatic effect.

A stable equilibrium may suddenly disappear, forcing the system towards a completely different state. Alternatively, a previously stable state may become unstable, allowing new patterns of behaviour to emerge.

These abrupt transitions are particularly important in biology because they frequently correspond to critical biological events.

Examples include

the commitment of a stem cell to a differentiated cell type,
the collapse of an ecological population,
the activation of developmental programs,
transitions between healthy and diseased physiological states.

Although the underlying parameters change continuously, the biological response may appear sudden and irreversible.

From equilibrium to bifurcation

Understanding these transitions requires extending equilibrium analysis.

Rather than asking whether an equilibrium is stable under fixed conditions, we now investigate how the equilibria themselves change as biological parameters vary.

This perspective introduces one of the central concepts of dynamical systems theory: the bifurcation.

A bifurcation occurs when a small change in a system parameter causes a qualitative change in the behaviour of the system. New equilibria may appear, existing equilibria may disappear, or stable states may become unstable.

Bifurcations therefore provide a mathematical explanation for biological switches, developmental decisions, and many other transitions between alternative functional states.

In the next chapter, we will explore how these qualitative changes arise and why they play such an important role in biological systems.

Key concepts

Equilibria depend on the biological parameters of a system.
Changes in environmental or physiological conditions can shift equilibrium states.
Small parameter changes may produce large qualitative changes in system behaviour.
Such qualitative transitions are called bifurcations.
Bifurcation theory provides a framework for understanding biological switches and state transitions.

Summary

Equilibrium analysis explains the long-term behaviour of biological systems under fixed conditions. Real biological systems, however, continuously experience changes in their environment and internal regulation. As these changes alter the underlying parameters of a system, the number, location, and stability of equilibrium points may also change. Understanding these qualitative transitions requires the study of bifurcations, which form the subject of the next chapter.

Self-check questions

Why are equilibrium points not fixed properties of a biological system?
Give three biological examples in which changing parameters alter the equilibrium state.
Why can a small parameter change sometimes produce a dramatic biological response?
What is meant by a bifurcation?
Why are bifurcations important for understanding biological decision making?