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

Suppose $r>1$ and set $q=\log r/(r-1)$. Substitution of the positive equilibrium into the Jacobian matrix givesPart (b) showed $f(x)<0$ whenever $x\ne 1$. In particular,gives $\log r<r-1$, whereasgives $\det J_{\ast }>1$. The latter already violates the Jury stability criterion condition $1-\det J_{\ast }>0$. More directly, the product of the two eigenvalues is greater than one, so at least one has modulus greater than one. Therefore the positive equilibrium is unstable. This is the instability of the Nicholson-Bailey equilibrium; parasitoid feedback in this simple model does not stabilize coexistence.