Skip to main content
SpringerLink
Log in

Relaxation oscillations in a system with delays modeling the predator–prey problem

  • Published:
Automatic Control and Computer Sciences Aims and scope Submit manuscript

Abstract

A new asymptotic method for investigating complex relaxation oscillations in a system with delay is offered. By applying it, we can reduce the problem of predator–prey system dynamics to that of one-dimensional maps analysis. Some conclusions of a biological nature based on asymptotic analysis are made.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Yang, K., Delay Differential Equations. With Applications in Population Dynamics, Academic Press, 1993.

    Google Scholar 

  2. Wright, E.M., A non-linear differential equation, J. Reine Angew. Math., 1955, vol. 194, p. 66–87.

    MathSciNet  MATH  Google Scholar 

  3. Kakutani, S. and Markus, L., On the non-linear difference-differential equation y(t) = (a–by(t–τ))y(t). Contributions to the theory of non-linear oscillations, Ann. Math. Stud., 1958, vol. 4, p. 1–18.

    MathSciNet  Google Scholar 

  4. Kaschenko, S.A., On the estimate of the global sustainability region of the Hutchinson equation in the parameter space, in Nelineynyye kolebaniya v zadachakh ekologii (Nonlinear Oscillations in Problems of Ecology), Yaroslavl: YarGU, 1985, p. 55–62.

    Google Scholar 

  5. Jones, G.S., The existence of periodic solutions of f(x) =–αf(x–1)[1 + f(x)], T. Math. Anal. Appl., 1962, vol. 5, p. 435–450.

    Article  MATH  Google Scholar 

  6. Kaschenko, S.A., The asymptotic of the periodic solution of the generalized Hutchinson equation, in Issledovaniya po ustoychivosti i teorii kolebaniy (Studies on the Stability and Oscillation Theory), Yaroslavl: YarGU, 1981, p. 64–85.

    Google Scholar 

  7. Kaschenko, S.A., Periodic solutions of the equation x(t) =–lx(t–1)[1 + x(t)], in Issledovaniya po ustoychivosti i teorii kolebaniy (Studies on the Stability and Oscillation Theory), Yaroslavl: YarGU, 1978, p. 110–117.

    Google Scholar 

  8. Glyzin, S.D., Kolesov, A.Yu., and Rozov N.Kh., Extremal dynamics of the generalized Hutchinson equation, Comput. Math. Math. Phys., 2009, vol. 49, no. 1, p. 71–83.

    Article  MathSciNet  Google Scholar 

  9. Kolesov, Yu.S., Mathematical models of ecology, in Issledovaniya po ustoychivosti i teorii kolebaniy (Studies on the Stability and Oscillation Theory), Yaroslavl, 1979, p. 3–40.

    Google Scholar 

  10. Kolesov, Yu.S. and Shvitra, D.I., Avtokolebaniya v sistemakh s zapazdyvaniyem (Self-Oscillations in Systems with Delay), Vilnius: Mokslas, 1979.

    Google Scholar 

  11. Kolesov, Yu.S. and Kubyshkin, E.P., The dual-frequency approach to the predator–prey problem, in Issledovaniya po ustoychivosti i teorii kolebaniy (Studies on the Stability and Oscillation Theory), Yaroslavl, 1979, p. 111–121.

    Google Scholar 

  12. Glyzin, S.D., On the stabilizing role of the non-homogeneous resistance of the environment in the predator–prey problem, in Issledovaniya po ustoychivosti i teorii kolebaniy (Studies on the Stability and Oscillation Theory), Yaroslavl, 1982, p. 126–129.

    Google Scholar 

  13. Zakharov, A.A., Kolesov, Yu.S., Spokoynov, A.N., and Fedotov, N.B., Theoretical explanation of the ten-year oscillation cycle in the number of mammals in Canada and Yakutia, in Issledovaniya po ustoychivosti i teorii kolebaniy (Studies on the Stability and Oscillation Theory), Yaroslavl, 1980, p. 79–131.

    Google Scholar 

  14. Kolesov, Yu.S. and Kubyshkin, E.P., Numerical investigation of a system of differential-difference equations modeling the predator–prey problem, in Faktory raznoobraziya v matematicheskoy ekologii i populyatsionnoy genetike (Diversity Factors in Mathematical Ecology and Population Genetics), Pushchino, 1980, p. 54–62.

    Google Scholar 

  15. Zakharov, A.A., Numerical study of the Kolesov system of equations modeling the predator–prey problem considering the predator’s pressure on the prey and its migration beyond the habitat, Differ. Uravn. Ikh Primen., 1981, no. 29, p. 9–26.

    MATH  Google Scholar 

  16. Glyzin, S.D., Consideration of age groups for the Hutchinson’s equation, Model. Anal. Inf. Syst., 2007, vol. 14, no. 3, p. 29–42.

    MathSciNet  Google Scholar 

  17. Kaschenko, S.A., Using the large parameter methods to research the system of nonlinear differential-difference equations modeling the predator–prey problem, Dokl. Akad. Nauk SSSR, 1982, vol. 266, p. 792–795.

    MathSciNet  Google Scholar 

  18. Kaschenko, S.A., Biological explanation of some of the laws of functioning of the simplest ecosystems in extreme cases, in Issledovaniya po ustoychivosti i teorii kolebaniy (Studies on the Stability and Oscillation Theory), Yaroslavl, 1982, p. 85–103.

    Google Scholar 

  19. Kaschenko, S.A., Periodic solutions of the system of nonlinear equations with delays modeling the predator–prey problem, in Issledovaniya po ustoychivosti i teorii kolebaniy (Studies on the Stability and Oscillation Theory), Yaroslavl, 1981, p. 136–143.

    Google Scholar 

  20. Kaschenko, S.A., Stationary modes in the predator–prey problem, Preprint of the Institute of Mathematics of the ANUkrSSR, Kiev, 1984.

    Google Scholar 

  21. Kaschenko, S.A., The asymptotic of solutions of the generalized Hutchinson equation, Model. Anal. Inf. Syst., 2012, vol. 19, no. 3, p. 32–62.

    Google Scholar 

  22. Sharkovskii, A.N., Maistrenko, Yu.L., and Romanenko, E.Yu., Difference Equations and Their Applications, Kluwer Academic Publishers, 1993.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Demidov Yaroslavl State University, ul. Sovetskaya 14, Yaroslavl, 150000, Russia

    S. A. Kaschenko

  2. National Research Nuclear University MEPhI, sh. Kashirskoye 31, Moscow, 115409, Russia

    S. A. Kaschenko

Authors
  1. S. A. Kaschenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. A. Kaschenko.

Additional information

Original Russian Text © S.A. Kaschenko, 2013, published in Modelirovanie i Analiz Informatsionnykh Sistem, 2013, No. 1, pp. 52–98.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kaschenko, S.A. Relaxation oscillations in a system with delays modeling the predator–prey problem. Aut. Control Comp. Sci. 49, 547–581 (2015). https://doi.org/10.3103/S0146411615070111

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0146411615070111

Keywords

Search

Navigation