NUMERICAL STUDY OF COEXISTENCE OF POPULATIONS IN AN ENVIRONMENTAL NICHE
Abstract
The predator-prey interactions on the spatial heterogeneous two-dimensional area are described. The model is written as a system of nonlinear parabolic equations for two closely related predator populations and two prey populations competing for the general resource. It is shown that under certain relationships between the parameters and the variable natural habitat resource functions, the model belongs to the class of the cosymmetric dynamical systems. In this case, there is a continuous family of stationary distributions of the coexistent populations. The simulation experiment is based on the method of straight lines, and on the scheme of staggered grids. The balance method is used for the approximation in spatial variables of the task on a rectangular area. The results showing the model capabilities for describing the formation of the population stationary distributions are presented. The formation of the biological structures is studied under the growth parameter heterogeneity; the conditions for the coexistence of closely related types are analyzed.
About the Authors
Alexander Vladimirovich BudyanskiyRussian Federation
Mikhail Gennadyevich Kruglikov
Russian Federation
Vyacheslav Georgiyevich Tsybulin
Russian Federation
References
1. Murray, J. D. Mathematical Biology II. Spatial models and Biomedical Applications / J. D. Murray. — Springer—Verlag, 2003. — 1082 p.
2. Гаузе, Г. Ф. Борьба за существование / Г. Ф. Гаузе. — Ижевск : Ин-т компьютерных ис-следований, 2002. — 234 с.
3. Бигон, М. Экология. Особи, популяции и сообщества / М. Бигон, Дж. Харпер, К. Таунсенд. — Москва : Мир, 1989. — 1144 с.
4. Белотелов, Н. В. Популяционные модели с нелинейной диффузией / Н. В. Белотелов, А. И. Лобанов // Математическое моделирование. — 1997. — Т. 9, № 12. — C. 43‒56.
5. Юдович, В. И. Косимметрия, вырождение решений операторных уравнений, возникно-вение фильтрационной конвекции / В. И. Юдович // Математические заметки. — 1991. — T. 49, № 5. — C. 142‒148.
6. Yudovich, V. I. Secondary cycle of equilibria in a system with cosymmetry, its creation by bi-furcation and impossibility of symmetric treatment of it. Chaos, 1995, vol. 5, no. 2, pp. 402‒411.
7. Govorukhin, V. Computer experiments with cosymmetric models. Z. Angew. Math. Mech, 1996, vol. 76, pp. 559‒562.
8. Banegje, M., Petrovski, S. Self-organised spatial patterns and chaos in a ratio-depended predator-prey system. J. Theor. Biol., 2011, vol. 4, pp. 37‒53.
9. Xue, L. Pattern formation in a predator-prey model with spatial effect. Physica A: Statistical mechanics and its applications, 2012, vol. 391, pp. 5987‒5996.
10. Будянский, А. В. Моделирование пространственно-временной миграции близкород-ственных популяций / А. В. Будянский, В. Г. Цибулин // Компьютерные исследования и моделиро-вание. — 2011. — Т. 3, № 4. — С. 477‒488.
11. Мишугова, Г. В. Моделирование процесса загрязнения атмосферы / Г. В. Мишугова // Вестник Дон. гос. техн. ун-та. — 2012. — № 8 (69). — С. 12‒17.
12. Заковортный, В. Л. Моделирование эволюции динамической системы, взаимодействующей со средой / В. Л. Заковоротный, Фам Дин Тунг // Вестник Дон. гос. техн. ун-та. — 2006. — T. 6, № 3 (30). — С. 184‒200.
13. Kovaleva, E. S., Frischmuth, K., Tsybulin, V. G. Dynamics of nonlinear parabolic equations with cosymmetry. Computer Algebra in Scientific Computing, CASC, 2007, pp. 265‒274.
14. Frischmuth, K., Kovaleva, E. S., Tsybulin, V. G. Family of equilibria in a population kinetics model and its collapse. Nonlinear Analysis: Real World Applications, 2011, vol. 12, pp. 145‒155.
Review
For citations:
Budyanskiy A.V., Kruglikov M.G., Tsybulin V.G. NUMERICAL STUDY OF COEXISTENCE OF POPULATIONS IN AN ENVIRONMENTAL NICHE. Vestnik of Don State Technical University. 2014;14(2):28-35. (In Russ.) https://doi.org/10.12737/4475