Simple mathematical models for cannibalism: A critique and a new approach (Q1072963)

We show how to incorporate a functional response in recent models of \textit{M. E. Gurtin} and \textit{D. S. Levine} [see e.g., SIAM J. Appl. Math. 42, 94-108 (1982; Zbl 0501.92021)] and others for egg cannibalism. Starting from a relatively complicated model with vulnerability spread over an age interval of finite duration \(\epsilon\), we arrive at a much simpler model by passing to the limit \(\epsilon\) \(\downarrow 0\). It turns out that survivorship through the vulnerable stage is implicitly determined by the solution of a scalar equation. Subsequently we study the existence and stability of steady states, and we find (analytically in a simple case, numerically in more general situations) curves in a two-dimensional parameter space where a nontrivial steady state loses its stability and a periodic solution arises through a Hopf bifurcation.