Adaptive behaviour and multiple equilibrium states in a predator–prey model

Abstract

There is evidence that multiple stable equilibrium states are possible in real-life ecological systems. Phenomenological mathematical models which exhibit such properties can be constructed rather straightforwardly. For instance, for a predator–prey system this result can be achieved through the use of non-monotonic functional response for the predator. However, while formal formulation of such a model is not a problem, the biological justification for such functional responses and models is usually inconclusive.In this note, we explore a conjecture that a multitude of equilibrium states can be caused by an adaptation of animal behaviour to changes of environmental conditions. In order to verify this hypothesis, we consider a simple predator–prey model, which is a straightforward extension of the classic Lotka–Volterra predator–prey model. In this model, we made an intuitively transparent assumption that the prey can change a mode of behaviour in response to the pressure of predation, choosing either “safe” of “risky” (or “business as usual”) behaviour. In order to avoid a situation where one of the modes gives an absolute advantage, we introduce the concept of the “cost of a policy” into the model. A simple conceptual two-dimensional predator–prey model, which is minimal with this property, and is not relying on odd functional responses, higher dimensionality or behaviour change for the predator, exhibits two stable co-existing equilibrium states with basins of attraction separated by a separatrix of a saddle point.