Ecological stability
Ecological stability concerns how an interacting biological system behaves after a disturbance. Several distinct meanings of stability are used, so the mathematical definition must be stated clearly.
Equilibrium stability
Suppose a system has an equilibrium \(\mathbf N^*\). It is locally stable if sufficiently small disturbances from that equilibrium decay and trajectories return toward it.
Linearisation
For a system
\[\frac{d\mathbf N}{dt}=\mathbf F(\mathbf N),\]local behaviour near an equilibrium can be studied using the Jacobian matrix
\[J=\left[\frac{\partial F_i}{\partial N_j}\right]_{\mathbf N=\mathbf N^*}.\]If all eigenvalues of \(J\) have negative real parts, the equilibrium is locally asymptotically stable.
Biological interpretation
Stability can also involve persistence, resistance to disturbance, recovery rate or variability. These concepts are related but not identical.
Key idea. Ecological stability connects species interactions to dynamical-systems theory. The interaction structure determines whether disturbances decay, persist or grow.