TY - JOUR

T1 - Adaptive behaviour and multiple equilibrium states in a predator-prey model

AU - Pimenov, Alexander

AU - Kelly, Thomas C.

AU - Korobeinikov, Andrei

AU - O'Callaghan, Michael J.A.

AU - Rachinskii, Dmitrii

PY - 2015/1/1

Y1 - 2015/1/1

N2 - © 2015 Elsevier Inc. 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.

AB - © 2015 Elsevier Inc. 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.

KW - Co-existence

KW - Fold bifurcation

KW - Multiple equilibria

KW - Predator pit

KW - Predator-prey model

KW - Stability of biosystems

U2 - 10.1016/j.tpb.2015.02.004

DO - 10.1016/j.tpb.2015.02.004

M3 - Article

VL - 101

SP - 24

EP - 30

JO - Theoretical Population Biology

JF - Theoretical Population Biology

SN - 0040-5809

ER -