 Stability threshold of a discrete exponential predator-prey model

For $n^{\prime }=rn{e}^{-p}$ and $p^{\prime }=n(1-b{e}^{-p})$, with $0<b<1<r$, the positive equilibrium is $p_c=\log r$, $n_c=r\log r/(r-b)$. Its Jacobian has trace $1+b{n}_c/r$ and determinant $n_c$. The unit-disc root conditions reduce to $n_c<1$. At $n_c=1$ the roots form a nonreal conjugate pair and cross the unit circle; their argument satisfies $2\cos \theta =1+b/r$. The unique threshold lies between one and $e$ because $n_c$ increases strictly and $\log r=1-b/r<1$ at threshold.