ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


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Kashchenko I. S., Kashchenko S. A. Dynamics of equation with two delays modelling the number of population. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, iss. 2, pp. 21-38. DOI: 10.18500/0869-6632-2019-27-2-21-38

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Language: 
Russian
Article type: 
Article
UDC: 
517.9

Dynamics of equation with two delays modelling the number of population

Autors: 
Kashchenko I. S., Yaroslavl State University
Kashchenko Sergej Aleksandrovich, Yaroslavl State University
Abstract: 

Issue. The paper investigates the behavior of solutions of a logistic equation with two delays from some neighborhood of the equilibrium state with a large value of the coefficient of linear growth. Such problems arise in modeling the population size taking into account the age structure, as a model of the number of insets, etc. Innovation. It is shown that the critical cases arising in the problem of the stability of an equilibrium state have infinite dimension: an infinitely large number of roots of the characteristic equation tend to the imaginary axis. In addition, in a number of studied situations, an additional degeneracy arises that significantly affects the structure of solutions. Investigation methods. To study the behavior of solutions in close to critical cases, an asymptotic method has been developed. With its help, special nonlinear equations – quasi-normal forms – whose solutions provide asymptotic approximations of solutions to the original problem. Results. It is shown that in critical cases the behavior of the solutions of the original singularly perturbed problem is determined by the dynamics of the quasi-normal form. The asymptotic formulas connecting their solutions are given. Complex parabolic Ginzburg–Landau equation can serve as a quasi-normal form, and for some degenerations, equations with one (possibly large) delay or the generalized Korteweg–de Vries equation. These tasks either do not contain a small parameter, or depend on it regularly. Conclusions. The behavior of solutions of a singularly perturbed logistic equation with two delays is studied. Critical cases are found and bifurcations are investigated. It is shown that the system under study has such dynamic effects as multistability and hypermultistability, as well as an infinite process of direct and inverse bifurcations as the small parameter tends to zero.

Acknowledgements. The reported study was funded by RFBR according to the research project No 18-29-10043.

Reference: 
  1. Hutchinson G.E. Circular causal in ecology. Ann. N.Y. Acad. Sci., 1948, vol. 50, pp. 221–246.
  2. Erneux T. Applied Delay Differential Equations. Berlin: Springer, 2009.
  3. Wright E.M. A non-linear difference-differential equation. Journal fur die Reine und Angewandte Mathematik,1955, vol. 194, pp. 66–87.
  4. Kakutani S., Markus L. On the non-linear difference-differential equation y ′ (t)=(a−by(t−τ))y(t). In: Contributions to the Theory of Nonlinear Oscillations; Ed. by Solomon Lefschetz. Princeton University Press, 1958. Vol. 4 of Annals of Mathematical Studies (AM-41), pp. 1– 18.
  5. Kashchenko S.A. Spatially heterogeneous structures in the simplest models with delay and diffusion. Mathematical Models and Computer Simulations, 1990, vol. 2, no. 9, pp. 49– 69. 
  6. Wu J. Theory and applications of partial functional differential equations. Applied mathematical sciences no. 119. Springer Verlag, 1996.
  7. May R.M. Stability and Complexity in Model Ecosystems. 2 edition. Princeton University Press, 2001.
  8. Kashchenko S.A. Asymptotics of the solutions of the generalized Hutchinson equation. Automatic Control and Computer Sciences, 2013, vol. 47, no. 7, pp. 470–494.
  9. Cushing J.M. Integrodifferential equations and delay models in population dynamics. Heidelberg, 1977.
  10. Kiseleva E.O. Local dynamics of the Hutchinson equation with two delays in the critical case of resonance 1:2. Modeling and Analysis of Information Systems, 2007, vol. 14, no. 2, pp. 53–57 (in Russian).
  11. Preobrazhenskaia M.M. Application of the method of quasi-normal forms to the mathematical model of a single neuron. Modeling and Analysis of Information Systems, 2014, vol. 21, no. 5, pp. 38–48 (in Russian).
  12. Kashchenko S.A. Dynamics of the logistic equation with two delays. Differential Equations, 2016, vol. 52, no. 5, pp. 538–548.
  13. May R.M. Time delays, density-dependence and single-species oscillations. Journal of Animal Ecology, 1974, vol. 43, no. 3, pp. 747–770.
  14. Kolesov Yu.S. Biophysics, 1983, vol. 28, no. 3, p. 513 (in Russian).
  15. Kashchenko S.A. Stationary modes of an equation describing fluctuations of an insect population. Soviet Physics. Doklady, 1983, vol. 28, no. 11, pp. 935–936.
  16. Glyzin S.D. Age groups in Hutchinson equations. Automatic Control and Computer Sciences, 2018, vol. 52, no. 7, pp. 714–727.
  17. Glyzin S.D., Kolesov A.Y., Rozov N.K. Extremal dynamics of the generalized Hutchinson equation. Computational Mathematics and Mathematical Physics, 2009, vol. 49, no. 1, pp. 71–83.
  18. Hale J., Sjoerd M.V.L. Introduction to Functional Differential Equations. New York: SpringerVerlag, 1993.
  19. Kaschenko S.A. Normalization techniques as applied to the investigation of dynamics of difference-differential equations with a small parameter multiplying the derivative. Differential equations, 1989, vol. 25, no. 8, pp. 1448–1451.
  20. Kashchenko I.S. Local dynamics of equations with large delay. Computational Mathematics and Mathematical Physics, 2008, vol. 48, no. 2, pp. 2172–2181.
  21. Kashchenko I.S., Kaschenko S.A. Applied nonlinear dynamics, 2008, vol. 16, no. 4, pp. 137–146 (in Russian).
  22. Wolfrum M., Yanchuk S. Eckhaus instability in systems with large delay. Physical Review Letters, 2006, vol. 96, 220201.
  23. Erneux T., Grasman J. Limit-cycle oscillators subject to a delayed feedback. Physical Review E., 2008, vol. 78, 026209.
  24. Yanchuk S., Perlikowski P. Delay and periodicity. Physical Review E., 2009, vol. 79, 046221.
  25. Balakin M.I., Ryskin N.M. Bifurcational mechanism of formation of developed multistability in a van der Pol oscillator with time-delayed feedback. Nelineinaya Dinamika, 2017, vol. 13, no. 2, pp. 151–164 (in Russian).
  26. Gaponov-Grekhov A.V., Rabinovich M.I. Ginzburg–Landau equation and nonlinear dynamics of nonequilibrium media. Radiophysics and Quantum Electronics, 1987, vol. 30, no. 2, pp. 93–102. 
  27. Akhromeeva T.S., Kurdyumov S.P., Malinetskii G.G., Samarskii A.A. Nonstationary Structures and Diffusion Chaos. Moscow: Nauka, 1992 (In Russian).
  28. Loskutov A.Y., Mikhailov A.S. Basic Theory of Complex Systems. Moscow; Izhevsk: Institute of Computer Science, 2007.
  29. Kashchenko A.A. Analysis of running waves stability in the Ginzburg–Landau equation with small diffusion. Automatic control and computer sciences, 2015, vol. 49, no. 11, pp. 514–517.
  30. Kashchenko S.A. A study of the stability of solutions of linear parabolic equations with nearly constant coefficients and small diffusion. Journal of Soviet Mathematics. 1992, vol. 60, no. 6, pp. 1742–1764.
  31. Lamb G.L.Jr. Elements of Soliton Theory. New York: Wiley-Interscience, 1980.
  32. Kudryashov N.A. Analytical Theory of Nonlinear Differential Equations. Moscow; Izhevsk: Institute of Computer Investigations, 2004.
  33. Zaitsev V.F., Polyanin A.D. Handbook of Exact Solutions for Ordinary Differential Equations. CRC press, 2002.
Received: 
18.12.2018
Accepted: 
01.03.2019
Published: 
24.04.2019
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