Stability in Models

The main idea

For \(X'=f(X)\), an equilibrium \(X^*\) satisfies \(f(X^*)=0\). Stability asks what nearby trajectories do.

1. Stable and unstable

Nearby solutions moving toward the equilibrium indicate local attraction; moving away indicates instability.

2. Sign analysis

In one dimension, the sign of \(f(X)\) on each side tells the direction of movement.

3. Derivative test

If \(f'(X^*)\lt0\), the equilibrium is locally asymptotically stable. If \(f'(X^*)\gt0\), it is unstable.

4. Logistic example

For \(f(P)=rP(1-P/K)\),

If \(r\gt0\), then \(f'(0)=r\gt0\), so \(0\) is unstable, while \(f'(K)=-r\lt0\), so \(K\) is stable.

5. Systems

For several variables, local stability is studied using the Jacobian matrix and its eigenvalues. Negative real parts of all eigenvalues imply local asymptotic stability under the usual linearisation conditions.

6. Local versus global

Local stability concerns small disturbances. Global stability concerns a larger set of admissible initial states.