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

Alexander Pimenov, Thomas C. Kelly, Andrei Korobeinikov, Michael J.A. O'Callaghan, Dmitrii Rachinskii

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    Abstract

    © 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.
    Original languageEnglish
    Pages (from-to)24-30
    JournalTheoretical Population Biology
    Volume101
    DOIs
    Publication statusPublished - 1 Jan 2015

    Keywords

    • Co-existence
    • Fold bifurcation
    • Multiple equilibria
    • Predator pit
    • Predator-prey model
    • Stability of biosystems

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  • Cite this

    Pimenov, A., Kelly, T. C., Korobeinikov, A., O'Callaghan, M. J. A., & Rachinskii, D. (2015). Adaptive behaviour and multiple equilibrium states in a predator-prey model. Theoretical Population Biology, 101, 24-30. https://doi.org/10.1016/j.tpb.2015.02.004