Adaptive behaviour and multiple equilibrium states in a predator-prey model.
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.
Weierstrass Institute, Mohrenstrasse 39, D-10117 Berlin, Germany. Electronic address: alexander.pimenov@wias-berlin.de.
Alexander
A
Pimenov
Department of Biology, Earth and Environmental Science, University College Cork, Ireland. Electronic address: t.kelly@ucc.ie.
Thomas C
TC
Kelly
Centre de Recerca Matemática, Campus de Bellaterra, Edifici C, 08193 Barcelona, Spain; Departament de Matemàtiques, Universitat Autònoma de Barcelona, Campus de Bellaterra, Edifici C, 08193 Barcelona, Spain. Electronic address: akorobeinikov@crm.cat.
Andrei
A
Korobeinikov
Department of Applied Mathematics, University College Cork, Ireland. Electronic address: mja.ocallaghan@ucc.ie.
Michael J A
MJ
O'Callaghan
Department of Mathematical Sciences, The University of Texas at Dallas, 800 W. Campbell Road, Richardson, TX 75080, USA. Electronic address: dmitry.rachinskiy@utdallas.edu.
Dmitrii
D
Rachinskii
D000223:Adaptation, Psychological; D000818:Animals; D001522:Behavior, Animal; D017753:Ecosystem; D004777:Environment; D020387:Food Chain; D008954:Models, Biological; D011157:Population Dynamics; D011235:Predatory Behavior; D012309:Risk-Taking
Co-existence; Fold bifurcation; Multiple equilibria; Predator pit; Predator–prey model; Stability of biosystems