Waves interaction in the Fisher–Kolmogorov equation with arguments deviation

Abstract

We considered the process of density wave propagation in the logistic equation with diffusion, such as Fisher–Kolmogorov equation, and arguments deviation. Firstly, we studied local properties of solutions corresponding to the considered equation with periodic boundary conditions using asymptotic methods. It was shown that increasing of period makes the spatial structure of stable solutions more complicated. Secondly, we performed numerical analysis. In particular, we considered the problem of propagating density waves interaction in infinite interval. Numerical analysis of the propagating waves interaction process, described by this equation, was performed at the computing cluster of YarSU with the usage of the parallel computing technology—OpenMP. Computations showed that a complex spatially inhomogeneous structure occurring in the interaction of waves can be explained by properties of the corresponding periodic boundary value problem solutions by increasing the spatial variable changes interval. Thus, the complication of the wave structure in this problem is associated with its space extension.

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

References

  1. 1.

    A. N. Kolmogorov, I. G. Petrovsky, and N. S. Piskunov, Bull. Univ. d’EtatMoscou, Ser. A 1, 1–26 (1937).

    Google Scholar 

  2. 2.

    R. A. Fisher, Ann. Eugen. 7, 355–369 (1937).

    Article  Google Scholar 

  3. 3.

    S. V. Aleshin, S. D. Glyzin, and S. A. Kaschenko, Model. Anal. Inform. Sist. 22, 304–321 (2015).

    Article  Google Scholar 

  4. 4.

    S. V. Aleshin, S. D. Glyzin, and S. A. Kaschenko, Model. Anal. Inform. Sist. 22, 609–628 (2015).

    Article  Google Scholar 

  5. 5.

    S. Kakutani and L. Markus, Ann. Math. Stud. 4, 1–18 (1958).

    Google Scholar 

  6. 6.

    S. A. Kaschenko, in Studies in Stability and the Theory of Oscillation (Yarosl. Gos. Univ., Yaroslavl’, 1981), Vol. 1, pp. 64–85 [in Russian].

    Google Scholar 

  7. 7.

    Yu. S. Kolesov, in Mathematical Models in Biology and Medicine (Inst. Mat. Kibernet., Vil’nyus, 1985), No. 1, pp. 93–103 [in Russian].

    Google Scholar 

  8. 8.

    Y. Kuang, Delay Differential Equations. With Applications in Population Dynamics (Academic, Boston, 1993).

    Google Scholar 

  9. 9.

    S. A. Kashchenko, Autom. Control Comput. Sci. 47, 470–494 (2013).

    Article  Google Scholar 

  10. 10.

    J. Wu, Theory and Applications of Partial Functional Differential Equations Theory and Applications of Partial Functional Differential Equations (Springer, New York, 1996).

    Google Scholar 

  11. 11.

    S. D. Glyzin, Model. Anal. Inform. Sist. 16 (3), 96–116 (2009).

    Google Scholar 

  12. 12.

    A. A. Kashchenko, Model. Anal. Inform. Sist. 18 (3), 58–62 (2015).

    MathSciNet  Google Scholar 

  13. 13.

    S. D. Glyzin, Autom. Control Comput. Sci. 47, 452–469 (2013).

    Article  Google Scholar 

  14. 14.

    S. D. Glyzin, A. Yu. Kolesov, and N. Kh. Rozov, Comput.Math. Math. Phys. 50, 816–830 (2010).

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to S. Aleshin.

Additional information

Submitted by A. M. Elizarov

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Aleshin, S., Glyzin, S. & Kaschenko, S. Waves interaction in the Fisher–Kolmogorov equation with arguments deviation. Lobachevskii J Math 38, 24–29 (2017). https://doi.org/10.1134/S199508021701005X

Download citation

Keywords and phrases

  • Fisher–Kolmogorov equation
  • diffusion
  • spatial deviation
  • delay differential equation
  • normal form
  • numerical analysis