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Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: Preisach operator ×

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Now showing 1 - 3 of 3
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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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    Stability results for a soil model with singular hysteretic hydrology
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Krejčí, Pavel; O'Kane, J. Philip; Pokrovskii, Alexei; Rachinskii, Dmitrii
    We consider a differential equation describing the mass balance in a soil hydrology model with noninvertible Preisach-type hysteresis. We approximate the singular Preisach operator by regular ones and show, as main result, that the solutions of the regularized problem converge to a solution of the original one as the regularization parameter tends to zero. For monotone right hand sides, we prove that the solution is unique. If in addition the external water sources are time periodic, then we find sufficient conditions for the existence, uniqueness, and asymptotic stability of periodic solutions.