Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 1 / B2 / f / Solution

The Jacobian matrix at the extinction fixed point iswith characteristic polynomialUnder $cr>d{b}^2$, $p(1)=d-cr/b^2<0$, whereas $p(q)\to \infty$ as $q\to \infty$. Hence its positive eigenvalueis greater than one. Its eigenvector can be taken with both components positive: choose $(r/(b^2{q}_{+}),1{)}^T$. Thus there are biologically admissible positive perturbations that grow. The extinction fixed point is unstable by linear stability analysis for a discrete dynamical system.