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Lyapunov Functionals and Stability of Stochastic Functional Differential Equations pp 257–282Cite as

Stability of Positive Equilibrium Point of Nonlinear System of Type of Predator–Prey with Aftereffect and Stochastic Perturbations

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Abstract

In Chap. 10 a system of two nonlinear differential equations is considered that is destined to unify different known mathematical models, in particular, very often investigated models of predator–prey type. In particular, a ratio-dependent predator–prey model is considered. The system under consideration is exposed to stochastic perturbations and is linearized in a neighborhood of the positive point of equilibrium. Two different ways of construction of asymptotic mean-square stability conditions are considered. The obtained asymptotic mean-square stability conditions for the trivial solution of the constructed linear system are at the same time sufficient conditions for the stability in probability of the positive equilibrium point of the initial nonlinear system under stochastic perturbations. The obtained stability regions are illustrated by six figures.

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Authors and Affiliations

  1. Department of Higher Mathematics, Donetsk State University of Management, Donetsk, Ukraine

    Leonid Shaikhet

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  1. Leonid Shaikhet
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Shaikhet, L. (2013). Stability of Positive Equilibrium Point of Nonlinear System of Type of Predator–Prey with Aftereffect and Stochastic Perturbations. In: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00101-2_10

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  • DOI: https://doi.org/10.1007/978-3-319-00101-2_10

  • Publisher Name: Springer, Heidelberg

  • Print ISBN: 978-3-319-00100-5

  • Online ISBN: 978-3-319-00101-2

  • eBook Packages: EngineeringEngineering (R0)

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