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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


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


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


References

1. Fisher R. A., “The Wave of Advance of Advantageous Genes”, Annals of Eugenics, 7 (1937), 355–369.

2. Колмогоров А.Н., Петровский И.Г., Пискунов Н.С., “Исследование уравнения диффузии, соединенной с возрастанием вещества, и его применение к одной биологической проблеме”, Сер. А. Математика и Механика, 1, 1937, 1–26; [French transl.: Kolmogorov A., Petrovsky I., Piscounov N., “Eґtude de l’eґquation de la diffusion avec croissance de la quantiteґ de matie`re et son application a` un proble`me biologique”, Moscou Univ. Bull. Math., 1:6 (1937), 1–25.]

3. Murray J. D., Mathematical Biology. I. An Introduction, Third Edition, Berlin, 2001.

4. Danilov V. G., Maslov V. P., Volosov K. A., Mathematical Modelling of Heat and Mass Transfer Processes, Kluwer, Dordrecht, 1995.

5. Volpert A., Volpert V., Volpert V., Traveling Wave Solutions of Parabolic Systems, American Mathematical Society, 2000.

6. Колесов Ю.С., “Математические модели экологии”, Исследования по устойчивости и теории колебаний, 1979, 3–40; [Kolesov Yu. S., “Matematicheskiye modeli ekologii”, Issledovaniya po ustoychivosti i teorii kolebaniy, 1979, 3–40, (in Russian).]

7. Колесов А.Ю., Колесов Ю.С., Релаксационные колебания в математических моделях экологии, Тр. МИАН, 199, Наука, М., 1993, 126 с.; [Kolesov A. Yu., Kolesov Yu. S., Relaxational oscillations in mathematical models of ecology, Trudy Mat. Inst. Steklov., 199, ed. E. F. Mishchenko, Nauka, Moscow, 1993, 126 pp., (in Russian).]

8. Гурли С.А., Соу Дж.В.-Х., Ву Дж.Х., “Нелокальные уравнения реакции-диффузии с запаздыванием: биологические модели и нелинейная динамика”, Современная математика. Фундаментальные направления, 1 (2003), 84–120; [English transl.: Gourley S. A., So J. W.-H., Wu J. H., “Nonlocality of Reaction-Diffusion Equations Induced by Delay: Biological Modeling and Nonlinear Dynamics”, Journal of Mathematical Sciences, 124:4 (2004), 5119–5153, (in Russian).]

9. Britton N. F., Reaction-diffusion equations and their applications to biology, Academic Press, New York, 1986.

10. Britton N. F., “Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model”, SIAM J. Appl. Math., 50 (1990), 1663–1688.

11. Алешин С.В., Глызин С.Д., Кащенко С.А., “Уравнение Колмогорова–Петровского–Пискунова с запаздыванием”, Моделирование и анализ информационных систем, 22:2 (2015), 304–321; [Aleshin S. V., Glyzin S. D., Kaschenko S. A., “Fisher–Kolmogorov–Petrovskii–Piscounov Equation with Delay”, Modeling and Analysis of Information Systems, 22:2 (2015), 304–321, (in Russian).]

12. Кащенко С.А., “Об установившихся режимах уравнения Хатчинсона с диффузией”, ДАН СССР, 292:2 (1987), 327–330; [Kashchenko S. A., “Ob ustanovivshihsja rezhimah uravnenija Hatchinsona s diffuziej”, DAN SSSR, 292:2 (1987), 327–330, (in Russian).]

13. Кащенко С.А., “Пространственно-неоднородные структуры в простейших моделях с запаздыванием и диффузией”, Математическое моделирование, 2:9 (1990), 49–69; [English transl.: Kashchenko S. A., “Spatial heterogeneous structures in the simplest models with delay and diffusion”, Matem. mod., 2:9 (1990), 49–69, (in Russian).]

14. Kashchenko S. A., “Asymptotics of the Solutions of the Generalized Hutchinson Equation”, Automatic Control and Computer Science, 47:7 (2013), 470–494.

15. Глызин С.Д., “Разностные аппроксимации уравнения טּреакция-диффузияя на отрезке”, Моделирование и анализ информационных систем, 16:3 (2009), 96–116; [Glyzin S. D., “Difference approximations of “reaction – diffusion” equation on a segment”, Modeling and Analysis of Information Systems, 16:3 (2009), 96–116, (in Russian).]

16. Wu J., Theory and Applications of Partial Functional Differential Equations Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.

17. Glyzin S. D., “Dimensional Characteristics of Diffusion Chaos”, Automatic Control and Computer Sciences, 47:7 (2013), 452–469.

18. Глызин С.Д., Колесов А.Ю., Розов Н.Х., “Конечномерные модели диффузионного хаоса”, Журнал вычислительной математики и математической физики, 50:5 (2010), 860–875; [English transl.: Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “Finite-dimensional models of diffusion chaos”, Computational Mathematics and Mathematical Physics, 50:5 (2010), 816–830, (in Russian).]

19. Schuster H. G., Deterministic Chaos: An Introduction, 3 edition, Wiley-VCH, 1995, 320 Pp.


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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ISSN 1818-1015 (Print)
ISSN 2313-5417 (Online)