Ecological stability

Ecological stability describes how a biological system responds to disturbance. The word stability has several mathematical and ecological meanings, so the model should state exactly which one is being used.

Core idea. Stability does not simply mean “nothing changes.” Different questions ask whether disturbances remain small, decay back to an equilibrium, remain bounded, or push the system into a different long-term state.

Start with an equilibrium

Consider

\[\frac{d\mathbf N}{dt}=\mathbf F(\mathbf N),\qquad \mathbf N=(N_1,\ldots,N_m)^T.\]

An equilibrium \(\mathbf N^*\) satisfies \(\mathbf F(\mathbf N^*)=\mathbf0\).

Stability and asymptotic stability are different

An equilibrium is locally stable in the Lyapunov sense if trajectories that start sufficiently close remain close after a small disturbance.

It is locally asymptotically stable if, in addition, nearby trajectories return toward the equilibrium as time increases.

Important distinction. “Stable” does not automatically mean “returns to equilibrium.” Return requires asymptotic stability.

Linearisation

Write \(\mathbf x=\mathbf N-\mathbf N^*\). Near the equilibrium,

\[\frac{d\mathbf x}{dt}\approx J\mathbf x,\qquad J=\left[\frac{\partial F_i}{\partial N_j}\right]_{\mathbf N=\mathbf N^*}.\]

The Jacobian gives the first-order response of population growth rates to small changes in the state.

Eigenvalue criterion

\[\boxed{\operatorname{Re}(\lambda_i)<0\quad\text{for every eigenvalue }\lambda_i},\]

then the equilibrium is locally asymptotically stable. If at least one eigenvalue has positive real part, it is unstable.

If one or more eigenvalues have zero real part and none have positive real part, linearisation may be inconclusive and nonlinear terms can determine stability.

Complex eigenvalues

For \(\lambda=a+bi\), the real part \(a\) controls exponential growth or decay and the imaginary part \(b\) produces oscillation. Negative real parts with nonzero imaginary parts therefore produce damped oscillations.

Local is not global

Local asymptotic stability concerns sufficiently small disturbances. A large disturbance can move the system outside the neighbourhood where the local analysis applies or into the basin of another attractor.

Basins and alternative stable states

The basin of attraction of an attractor is the set of initial states whose trajectories approach it. If several attractors exist, crossing a basin boundary can produce a different long-term outcome or regime shift.

Resistance and resilience

Resistance describes how little a system changes under disturbance. Resilience often refers to how rapidly it returns, although terminology varies across ecology.

Near a locally asymptotically stable equilibrium, the eigenvalue whose real part is closest to zero often controls the slowest local recovery rate.

Persistence and variability

Persistence asks whether species remain present rather than becoming extinct. Variability describes the size of population fluctuations. These are distinct from local equilibrium stability.

Stable cycles

Not every stable long-term behaviour is a fixed point. A system may have a stable limit cycle, where nearby trajectories approach a repeating orbit instead.

Food-web stability

In a multispecies system, the Jacobian combines self-regulation and cross-species effects. Stability depends on interaction signs, magnitudes, network structure and the equilibrium at which the matrix is evaluated.

Critical slowing down

As some systems approach a bifurcation, the dominant recovery rate can approach zero from below. Disturbances then decay more slowly. This is called critical slowing down.

Early-warning indicators are not universal guarantees. Their usefulness depends on the mechanism, data quality and noise structure.

Practical workflow

Find biologically feasible equilibria, calculate the Jacobian, evaluate it at each equilibrium, compute its eigenvalues, interpret their real parts, and then decide whether nonlinear, global or stochastic analysis is also needed.

Key idea. Local stability and local asymptotic stability are not identical. Negative real parts of all Jacobian eigenvalues imply local asymptotic stability, while resistance, resilience, persistence and alternative stable states describe different ecological aspects of disturbance response.