Local Dynamics of Logistic Equation with Delay and Diffusion
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 c...
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| Format: | Article |
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MDPI AG
2021-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/13/1566 |
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| author | Sergey Kashchenko |
| author_facet | Sergey Kashchenko |
| author_sort | Sergey Kashchenko |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-10T09:51:01Z |
| format | Article |
| id | doaj.art-a005e166830147a28ecb84ff4c70b3e8 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T09:51:01Z |
| publishDate | 2021-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-a005e166830147a28ecb84ff4c70b3e82023-11-22T02:46:23ZengMDPI AGMathematics2227-73902021-07-01913156610.3390/math9131566Local Dynamics of Logistic Equation with Delay and DiffusionSergey Kashchenko0Centre of Integrable Systems, P. G. Demidov State University, 150003 Yaroslavl, RussiaThe 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.https://www.mdpi.com/2227-7390/9/13/1566logistic equationdiffusiondynamicsstabilitybifurcationsasymptotics |
| spellingShingle | Sergey Kashchenko Local Dynamics of Logistic Equation with Delay and Diffusion Mathematics logistic equation diffusion dynamics stability bifurcations asymptotics |
| title | Local Dynamics of Logistic Equation with Delay and Diffusion |
| title_full | Local Dynamics of Logistic Equation with Delay and Diffusion |
| title_fullStr | Local Dynamics of Logistic Equation with Delay and Diffusion |
| title_full_unstemmed | Local Dynamics of Logistic Equation with Delay and Diffusion |
| title_short | Local Dynamics of Logistic Equation with Delay and Diffusion |
| title_sort | local dynamics of logistic equation with delay and diffusion |
| topic | logistic equation diffusion dynamics stability bifurcations asymptotics |
| url | https://www.mdpi.com/2227-7390/9/13/1566 |
| work_keys_str_mv | AT sergeykashchenko localdynamicsoflogisticequationwithdelayanddiffusion |