Bistability in a size-structured population model of cannibalistic fish--a continuation study.

By numerical continuation of equilibria, we study a size-structured model for the dynamics of a cannibalistic fish population and its alternative resource. Because we model the cannibalistic interaction as dependent on the ratio of cannibal length and victim length, a cannibal experiences a size distribution of potential victims which depends on its own body size. We show how equilibria of the resulting infinite-dimensional dynamical system can be traced with an existing method for numerical continuation for physiologically structured population models. With this approach we found that cannibalism can induce bistability associated with a fold (or, saddle-node) bifurcation. The two stable states can be qualified as 'stunted' and 'piscivorous', respectively. We identify a new ecological mechanism for bistability, in which the energy gain from cannibalism plays a crucial role: Whereas in the stunted population state cannibals consume their victims, on average, while they are very small and yield little energy, in the piscivorous state cannibals consume their victims not before they have become much bigger, which results in a much higher mean yield of cannibalism. We refer to this mechanism as the 'Hansel and Gretel' effect. It is not related to any individual 'choice' or 'strategy', but depends purely on a difference in population size distribution. We argue that studying dynamics of size-structured population models with this new approach of equilibrium continuation extends the insight that can be gleaned from numerical simulations of the model dynamics.