Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2025 / Paper 1 / 5 / f

The Jacobian matrix of the update map isIts eigenvalues are $r$ and zero. The extinction equilibrium is locally asymptotically stable for $0<r<1$ and unstable for $r>1$, by linear stability analysis for a discrete dynamical system. In the latter case, even a small perturbation along $P=0$ grows as $H_t=r^t{H}_0$.

At $r=1$, an eigenvalue equals one, so the strict linear criterion does not decide stability. Directly, $0\le H_{t+1}\le H_t$ and $0\le P_{t+1}\le b{H}_t$. Starting sufficiently close to extinction therefore keeps both coordinates close for all time, proving Lyapunov stability in the nonnegative state space. It is not asymptotically stable, since every nearby $(H,0)$ is another equilibrium. This distinction makes the statement “stable when $r>1$ fails” precise.