Dynamical Properties of the Fisher–Kolmogorov–Petrovskii–Piscounov Equation with Deviation of the Spatial Variable
https://doi.org/10.18255/1818-1015-2015-5-609-628
Abstract
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.
About the Authors
S. V. Aleshin
Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia
Scientific Center in Chernogolovka RAS, Lesnaya str., 9, Chernogolovka, Moscow region, 142432, Russia
Russian Federation
Russian Federation
S. D. Glyzin
Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia
Scientific Center in Chernogolovka RAS, Lesnaya str., 9, Chernogolovka, Moscow region, 142432, Russia
Russian Federation
Doctor, Professor
Russian Federation
Doctor, Professor
S. A. Kaschenko
Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150000, Russia
National Research Nuclear University MEPhI, Kashirskoye shosse 31, Moscow, 115409, Russia
Russian Federation
Doctor, Professor
Russian Federation
Doctor, Professor
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Review
For citations:
Aleshin S.V., Glyzin S.D., Kaschenko S.A. Dynamical Properties of the Fisher–Kolmogorov–Petrovskii–Piscounov Equation with Deviation of the Spatial Variable. Modeling and Analysis of Information Systems. 2015;22(5):609-628. (In Russ.) https://doi.org/10.18255/1818-1015-2015-5-609-628
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