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Type: Report × Subject: predator-prey model ×

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    Adaptive behaviour in a predator-prey model leads to multiple equilibrium states
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Pimenov, Alexander; Korobeinikov, Andrei; Rachinskii, Dmitrii
    There is evidence that multiple stable equilibrium states are possible in real-life ecological systems. In order to verify a hypothesis that such a multitude of equilibrium states can be caused by adapting of animal behaviour to changes of environmental conditions, we consider a simple predator-prey model where prey changes a mode of behaviour in response to the pressure of predation. This model exhibits two stable coexisting equilibrium states with basins of attraction separated by a separatrix of a saddle point.
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    Memory and adaptive behaviour in population dynamics: Anti-predator behaviour as a case study
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Pimenov, Alexander; Kelly, Thomas C.; Korobeinikov, Andrei; OCallaghan, Michael J.; Rachinskii, Dmitrii
    Memory enables to forecast future on the basis of experience, and thus, in some form, is principally important for the development of flexible adaptive behaviour by animal communities. To model memory, in this paper we use the concept of hysteresis, which mathematically is described by the Preisach operator. As case study, we consider anti-predator adaptation in the classic Lotka-Volterra predator-prey model. Despite its simplicity, the model allows to naturally incorporate essential features of an adaptive system and memory. Our analysis and simulations show that a system with memory can have a continuum of equilibrium states with non-trivial stability properties.
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    Homoclinic orbits in a two-patch predator-prey model with Preisach hysteresis operator
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Pimenov, Alexander; Rachinskii, Dmitrii
    Systems of operator-differential equations which hysteresis operators can have unstable equilibrium points with an open basin of attraction. In this paper, a numerical example of a robust homoclinic loop is presented for the first time in a population dynamics model with hysteretic response of prey to variations of predator. A mechanism creating this homoclinic trajectory is discussed.
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    Robust homoclinic orbits in planar systems with Preisach hysteresis operator
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Pimenov, Alexander; Rachinskii, Dmitrii
    We construct examples of robust homoclinic orbits for systems of ordinary differential equations coupled with the Preisach hysteresis operator. Existence of such orbits is demonstrated for the first time. We discuss a generic mechanism that creates robust homoclinic orbits and a method for finding them. An example of a homoclinic orbit in a population dynamics model with hysteretic response of the prey to variations of the predator is studied numerically.