= Solution
The <Jacobian matrix> of the update map is
$$
J(H,P)=\begin{pmatrix}
r e^{-aP}&-arHe^{-aP}\\
b(1-e^{-aP})&abHe^{-aP}
\end{pmatrix},\qquad
J(0,0)=\begin{pmatrix}r&0\\0&0\end{pmatrix}.
$$
Its <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 $H_t=r^tH_0$.
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 $0\le P_{t+1}\le bH_t$. 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.
Back to article page
