Abstract
We studied the dynamics of finite-difference approximation with respect to spatial variables of a logistic equation with delay and diffusion. It was assumed that the diffusion coefficient is small and the Malthusian coefficient is large. The question concerning the existence and asymptotic behavior of the attractors was studied using special asymptotic methods. It has been shown that there is a rich set of attractors of different types in the phase space: guiding centers, systems of helicon waves, etc. The major asymptotic characteristics of all the solutions from the corresponding attractors are adduced in this work. Typical graphics of the motion of wave fronts of different structures are represented in the paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Diekmann, O., Run for your life, a note on the asymptotic speed of propagation of an epidemic, J. Differ. Eq., 1979, vol. 33, pp. 58–73.
Kuang, Y., Delay differential equations with applications in population dynamics, in Mathematics in Science and Engineering, New York: Academic, 1993.
May, R.M., Stability and Complexity in Model Ecosystems, Princeton: Princeton University, 1975.
Murray, J.D., Mathematical Biology, Berlin: Springer-Verlag, 1993.
Wu, J.H., Theory and applications of partial functional differential equations, in Appl. Math. Sci., New York: Springer-Verlag, 1996.
Wu, J.H. and Zou, X.F., Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Differ. Eq., 2001, vol. 13, pp. 651–687.
Zou, X.F. and Wu, J.H., Existence of traveling wave fronts in delay reaction-diffusion system via monotone iteration method, Proc. Am. Math. Soc., 1997, vol. 125, pp. 2589–2598.
Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics, Dordrecht: Kluwer, 1992.
Gourley, S.A., So, J.W.-H., and Wu, J.H., Nonlocality of reaction-diffusion equations induced by delay: Biological modeling and nonlinear dynamics, J. Math. Sci. 2004, vol. 124, pp. 5119–5153.
Kolmogorov, A., Petrovskii, I., and Piskunov, N., An investigation of the diffusion equation combined with an increase in mass and its application to a biological problem, Bull. Univ. Moscow Ser. Int. A 1937, vol. 1, pp. 1–26.
Kolesov, Yu.S. and Maiorov, V.V., Space and time self-organization in one-species biocenosis, in Dinamika biologicheskih populjacij: Mejvuz. sb. (Dynamics of Biological Populations. Interinst. Coll. Papers), Gorkii, 1986, pp. 3–13.
Kolesov, Yu.S., Mathematical models in ecology, in Issledovanija po ustojchivosti i teorii kolebanij (Studies on Stability and Oscillation Theory), Yaroslavl, 1979, pp. 3–40.
Svirejev, Yu. M., Nelinejnye volny, dissipativnye struktury i katastrofy v ekologii (Nonlinear Waves, Dissipative Structures and Catastrophes in Ecology), Moscow: Nauka, 1987.
Kashchenko, S.A., Spatial heterogeneous structures in the simplest models with delay and diffusion, Matem. Model, 1990, vol. 2, pp. 49–69.
Goryachenko, V.D. and Kolchin, V.A., To dynamics of number of distinct population taking into account delay in reproduction, in Nelinejnye kolebanija v zadachah ekologii: Mejvuz. sb. (Nonlinear Oscillations in Problems of Ecology. InterInst. Coll. Papers), Yaroslavl, 1985, pp. 23–44.
Goryachenko, V.D. and Zolotarev, S.L., Study of two-point model of population number dynamics, in Dinamika biologicheskih populjacij. Mejvuz. Sb. (Dynamics of Biological Population. Interinstitute Coll. Papers), Gorkii, 1986, pp. 24–31.
Glyzin, S.D., Kolesov, A.Yu., and Rozov, N.Kh., Finite-dimensional models of diffusion chaos, Compt. Mathem. Mathem. Phys., 2010, vol. 50, pp. 816–830.
Glyzin, S.D., Dimensional characteristics of diffusion chaos, Model. Anal. Inform. Sist., 2013, vol. 20, no. 1, pp. 30–51.
Kashchenko, S.A., Study of system of nonlinear differential-difference equations modeling the “beast of preysacrifice” problem, Doklady Akad. Nauk SSSR, 1982, vol. 266, pp. 792–795.
Kashchenko, S.A., On steady regimes of Hatchinson’s equation with diffusion, Doklady Akad. Nauk SSSR, 1987, vol. 292, pp. 327–330.
Kolesov, Yu.S., On stability of space-homogeneous cycle in Hatchinson’s equation with diffusion, in Matem. modeli v biologii i medicine (Mathematical Models in Biology and Medicine), Vilnus: Inst. Math. Akad. Nauk Lit. SSR, no. 1, pp. 93–103.
Bestehorn, M., Grigorieva, E.V., Haken, H., and Kaschenko, S.A., Order parameters for class-B lasers with a long time delayed feedback, Physica D, 2000, vol. 145, pp. 111–129.
Glyzin, S.D., Difference approximations of “reaction-diffusion” equation on a segment, Model. Anal. Inform. Sist., 2009, vol. 16, no. 3, pp. 96–116.
Glyzin, S.D., Dynamic properties of the simplest finite-difference approximations of the “reaction-diffusion” boundary value problem, Differ. Eq., 1997, vol. 33, pp. 808–814.
Vladimirov, V.V. et al., Upravlenie riskom (Control of Risk), Moscow: Nauka, 2000.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © S.A. Kashchenko, V.E. Frolov, 2014 published in Modelirovanie i Analiz Informatsionnykh Sistem, 2014, No. 1, pp. 94–114.
About this article
Cite this article
Kashchenko, S.A., Frolov, V.E. Asymptotics of solutions of finite-difference approximations of a logistic equation with delay and small diffusion. Aut. Control Comp. Sci. 48, 502–515 (2014). https://doi.org/10.3103/S0146411614070104
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0146411614070104