Equilibria
An equilibrium is a state at which the model has no instantaneous tendency to change. If the system starts exactly there and the parameters remain fixed, it stays there.
One-dimensional systems
For
\[\frac{dx}{dt}=f(x),\]an equilibrium \(x^*\) satisfies
\[\boxed{f(x^*)=0}.\]This means the instantaneous rate of change is zero at \(x=x^*\).
Why setting the derivative to zero works
The derivative tells us the direction and speed of change. If \(dx/dt>0\), the state increases. If \(dx/dt<0\), the state decreases. At equilibrium, neither happens, so \(dx/dt=0\).
A simple example
Consider
\[\frac{dx}{dt}=3-x.\]Setting the right-hand side equal to zero gives
\[3-x=0,\]so
\[\boxed{x^*=3}.\]Equilibria are states, not trajectories
The equilibrium value \(x^*=3\) is one special state. A trajectory starting at \(x=1\) is not an equilibrium, even if it later approaches 3.
Biological example: logistic growth
For
\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right),\]equilibria satisfy
\[rN\left(1-\frac{N}{K}\right)=0,\]so
\[\boxed{N^*=0\qquad\text{or}\qquad N^*=K}.\]The first equilibrium represents extinction. The second represents the carrying-capacity state of this model.
Biological feasibility matters
Mathematical equations can produce equilibrium values that are impossible biologically. For example, a negative population equilibrium may be algebraically valid but biologically meaningless.
Equilibria in systems of equations
For
\[\frac{d\mathbf x}{dt}=\mathbf F(\mathbf x),\]an equilibrium \(\mathbf x^*\) satisfies
\[\boxed{\mathbf F(\mathbf x^*)=\mathbf0}.\]Every component derivative must be zero at the same time.
Nullclines
For
\[\frac{dx}{dt}=f(x,y),\qquad \frac{dy}{dt}=g(x,y),\]the \(x\)-nullcline is \(f(x,y)=0\) and the \(y\)-nullcline is \(g(x,y)=0\). Equilibria occur at their intersections.
Boundary and interior equilibria
| Type | Meaning |
|---|---|
| boundary equilibrium | one or more populations are zero |
| interior equilibrium | all populations are positive |
| disease-free equilibrium | infected compartments are zero |
| endemic equilibrium | infection remains present at a positive level |
| coexistence equilibrium | multiple species remain positive |
Existence can depend on parameters
Some equilibria exist only when parameters satisfy certain conditions. An endemic equilibrium, for example, may require a transmission parameter to exceed a threshold before the equilibrium becomes positive and biologically feasible.
Equilibrium does not mean no activity
A biological equilibrium can contain ongoing events. Births and deaths may both occur, infections and recoveries may continue, or individuals may move between compartments.
What is zero is the net rate of change, not necessarily every underlying process.
Equilibrium versus long-term behaviour
A system does not have to approach an equilibrium. It may oscillate, approach a limit cycle, behave chaotically, or leave the biologically relevant region.
Finding equilibria is therefore an important first step, not a complete description of the dynamics.
Equilibrium versus stability
Once an equilibrium has been found, the next question is what happens after a small disturbance.
If nearby trajectories remain close, the equilibrium is locally stable. If nearby trajectories also return toward the equilibrium, it is locally asymptotically stable. If nearby trajectories move away, it is unstable.
Why equilibria matter in biology
Equilibria can represent persistent population levels, extinction states, coexistence, disease-free states, endemic infection, chemical concentrations, tumour states or other long-term configurations.
They provide reference states around which stability, bifurcations, intervention thresholds and stochastic fluctuations can be analysed.
A practical workflow
Set every derivative equal to zero, solve the resulting algebraic equations, list all mathematical solutions, remove biologically infeasible ones, interpret the remaining equilibria, and then analyse their stability.