Instability of the Nicholson-Bailey equilibrium ($\det J_{\ast }=r\log r/(r-1)>1$)

At the positive Nicholson-Bailey model equilibrium, $\mathrm{tr}{J}_{\ast }=1+\log r/(r-1)$ lies between one and two, and the displayed determinant exceeds one. These follow from $(r-1)/r<\log r<r-1$ for $r>1$. The discriminant is therefore negative, and the conjugate eigenvalues have moduli greater than one. Nearby annual samples spiral outward rather than approach coexistence.