Abstract
In the paper we consider the problem of searching for coexisting modes in a nonlinear boundary value problem with a delay from population dynamics. For this we construct the asymptotic of spatially homogeneous cycle using the normal forms method and research the dependence of its stability on the diffusion parameter. Then we find coexisting attractors of the problem using numerical methods. Numerical experiment required an application of massively parallel computing systems and adaptation of solutions search algorithms to them. Based on the numerical analysis we come to the conclusion of the existence in the boundary value problem of solutions of two types. The first type has a simple spatial distribution and inherits the properties of a homogeneous solution. The second called the mode of self-organization is more complex distributed in space and is much more preferred in terms of population dynamics.
Similar content being viewed by others
References
A. Yu. Kolesov and V. V. Mayorov, “Spatial and temporal self-organization in single-species biocenosis,” in The Dynamics of Biological Populations, Interschool Collection of Articles (1986), pp. 3–13 [in Russian].
A. Yu. Kolesov and Yu. S. Kolesov, Relaxational Oscillations in Mathematical Models of Ecology, Vol. 199 of Proc. Steklov Inst. Math. (Am. Math. Soc., Providence, 1995).
S. A. Gourley, J. W.-H. So, and J. H. Wu, “Nonlocality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics,” J. Math. Sci. 124, 5119–5153 (2004).
N. F. Britton, Reaction-Diffusion Equations and their Applications to Biology (Academic, New York, 1986).
N. F. Britton, “Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model,” SIAM J. Appl.Math. 50, 1663–1688 (1990).
Yu. S. Kolesov and D. I. Svitra, Self-Oscillations in Systems with Delay (Mokslas, Vilnius, 1979) [in Russian].
S. A. Kashchenko, “Asymptotics of periodical solution of Hutchinson generalized equation,” Autom. Control Comput. Sci. 47, 470–494 (2013).
Y. Kuang, Delay Differential Equations. With Applications in Population Dynamics (Academic New York, 1993).
J. Wu, Theory and Applications of Partial Functional Differential Equations (Springer, New York, 1996).
E. M. Wright, “A non-linear difference-differential equation,” J. Reine Angew.Math. 194, 66–87 (1955).
A. Yu. Kolesov, “On stability of a spatially homogeneous cycle in Hutchinson equation with diffusion,” Mat. Modeli Biol.Med. 1, 93–103 (1985).
S. D. Glyzin, “Dimensional characteristics of diffusion chaos,” Autom. Control Comput. Sci. 47, 452–469 (2013).
S. D. Glyzin, A. Yu. Kolesov, and N. Kh. Rozov, “Finite-dimensional models of diffusion chaos,” Comput. Math. Math. Phys. 50, 816–830 (2010).
S. D. Glyzin, “Difference approximations of’ reaction–diffusion’ equation on a segment,” Model. Anal. Inform. Sistem 16 (3), 96–116 (2009).
Author information
Authors and Affiliations
Corresponding author
Additional information
Submitted by A. M. Elizarov
Rights and permissions
About this article
Cite this article
Glyzin, S., Goryunov, V. & Kolesov, A. Spatially inhomogeneous modes of logistic differential equation with delay and small diffusion in a flat area. Lobachevskii J Math 38, 898–905 (2017). https://doi.org/10.1134/S1995080217050110
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080217050110