Dynamical Properties of the Fisher–Kolmogorov–Petrovskii–Piscounov Equation with Deviation of the Spatial Variable
We consider the problem of density wave propagation of a logistic equation with deviation of the spatial variable and diffusion (Fisher-Kolmogorov equation with deviation of the spatial variable). A Ginzburg–Landau equation was constructed in order to study the qualitative behavior of the solution ne...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2015-10-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/281 |
| _version_ | 1826558973826826240 |
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| author | S. V. Aleshin S. D. Glyzin S. A. Kaschenko |
| author_facet | S. V. Aleshin S. D. Glyzin S. A. Kaschenko |
| author_sort | S. V. Aleshin |
| collection | DOAJ |
| description | We consider the problem of density wave propagation of a logistic equation with deviation of the spatial variable and diffusion (Fisher-Kolmogorov equation with deviation of the spatial variable). A Ginzburg–Landau equation was constructed in order to study the qualitative behavior of the solution near the equilibrium state. We analyzed the profile of the wave equation and found conditions for the appearance of oscillatory regimes. The numerical analysis of wave propagation shows that for a suficiently small spatial deviation this equation has a solution similar to the solution of the classical Fisher–Kolmogorov equation. The spatial deviation increasing leads to the existence of the oscillatory component in the spatial distribution of solutions. A further increase of the spatial deviation leads to destruction of the traveling wave. That is expressed in the fact that undamped spatio-temporal fluctuations exist in a neighborhood of the initial perturbation. These fluctuations are close to the solution of the corresponding boundary value problem with periodic boundary conditions. Finally, when the spatial deviation is suficiently large we observe intensive spatio-temporal fluctuations in the whole area of wave propagation. |
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| id | doaj.art-830047026ca84c91b3e2f770b62140ff |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2025-03-14T08:53:00Z |
| publishDate | 2015-10-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-830047026ca84c91b3e2f770b62140ff2025-03-02T12:46:57ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172015-10-0122560962810.18255/1818-1015-2015-5-609-628262Dynamical Properties of the Fisher–Kolmogorov–Petrovskii–Piscounov Equation with Deviation of the Spatial VariableS. V. Aleshin0S. D. Glyzin1S. A. Kaschenko2Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia Scientific Center in Chernogolovka RAS, Lesnaya str., 9, Chernogolovka, Moscow region, 142432, RussiaYaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia Scientific Center in Chernogolovka RAS, Lesnaya str., 9, Chernogolovka, Moscow region, 142432, RussiaYaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia National Research Nuclear University MEPhI, Kashirskoye shosse 31, Moscow, 115409, RussiaWe consider the problem of density wave propagation of a logistic equation with deviation of the spatial variable and diffusion (Fisher-Kolmogorov equation with deviation of the spatial variable). A Ginzburg–Landau equation was constructed in order to study the qualitative behavior of the solution near the equilibrium state. We analyzed the profile of the wave equation and found conditions for the appearance of oscillatory regimes. The numerical analysis of wave propagation shows that for a suficiently small spatial deviation this equation has a solution similar to the solution of the classical Fisher–Kolmogorov equation. The spatial deviation increasing leads to the existence of the oscillatory component in the spatial distribution of solutions. A further increase of the spatial deviation leads to destruction of the traveling wave. That is expressed in the fact that undamped spatio-temporal fluctuations exist in a neighborhood of the initial perturbation. These fluctuations are close to the solution of the corresponding boundary value problem with periodic boundary conditions. Finally, when the spatial deviation is suficiently large we observe intensive spatio-temporal fluctuations in the whole area of wave propagation.https://www.mais-journal.ru/jour/article/view/281attractorbifurcationfisher-kolmogorov equationginzburg-landau equation |
| spellingShingle | S. V. Aleshin S. D. Glyzin S. A. Kaschenko Dynamical Properties of the Fisher–Kolmogorov–Petrovskii–Piscounov Equation with Deviation of the Spatial Variable Моделирование и анализ информационных систем attractor bifurcation fisher-kolmogorov equation ginzburg-landau equation |
| title | Dynamical Properties of the Fisher–Kolmogorov–Petrovskii–Piscounov Equation with Deviation of the Spatial Variable |
| title_full | Dynamical Properties of the Fisher–Kolmogorov–Petrovskii–Piscounov Equation with Deviation of the Spatial Variable |
| title_fullStr | Dynamical Properties of the Fisher–Kolmogorov–Petrovskii–Piscounov Equation with Deviation of the Spatial Variable |
| title_full_unstemmed | Dynamical Properties of the Fisher–Kolmogorov–Petrovskii–Piscounov Equation with Deviation of the Spatial Variable |
| title_short | Dynamical Properties of the Fisher–Kolmogorov–Petrovskii–Piscounov Equation with Deviation of the Spatial Variable |
| title_sort | dynamical properties of the fisher kolmogorov petrovskii piscounov equation with deviation of the spatial variable |
| topic | attractor bifurcation fisher-kolmogorov equation ginzburg-landau equation |
| url | https://www.mais-journal.ru/jour/article/view/281 |
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