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Local Dynamics of Logistic Equation with Delay and Diffusion

Author

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  • Sergey Kashchenko

    (Centre of Integrable Systems, P. G. Demidov State University, 150003 Yaroslavl, Russia)

Abstract

The behavior of all the solutions of the logistic equation with delay and diffusion in a sufficiently small positive neighborhood of the equilibrium state is studied. It is assumed that the Andronov–Hopf bifurcation conditions are met for the coefficients of the problem. Small perturbations of all coefficients are considered, including the delay coefficient and the coefficients of the boundary conditions. The conditions are studied when these perturbations depend on the spatial variable and when they are time-periodic functions. Equations on the central manifold are constructed as the main results. Their nonlocal dynamics determines the behavior of all the solutions of the original boundary value problem in a sufficiently small neighborhood of the equilibrium state. The ability to control the dynamics of the original problem using the phase change in the perturbing force is set. The numerical and analytical results regarding the dynamics of the system with parametric perturbation are obtained. The asymptotic formulas for the solutions of the original boundary value problem are given.

Suggested Citation

  • Sergey Kashchenko, 2021. "Local Dynamics of Logistic Equation with Delay and Diffusion," Mathematics, MDPI, vol. 9(13), pages 1-18, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:13:p:1566-:d:588180
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    Cited by:

    1. Sergey Kashchenko, 2022. "Infinite–Dimensional Bifurcations in Spatially Distributed Delay Logistic Equation," Mathematics, MDPI, vol. 10(5), pages 1-32, February.
    2. Sergey Kashchenko, 2022. "Quasinormal Forms for Chains of Coupled Logistic Equations with Delay," Mathematics, MDPI, vol. 10(15), pages 1-32, July.

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