Relaxation oscillations in a system with delays modeling the predator–prey problem (original) (raw)

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