Waves interaction in the Fisher–Kolmogorov equation with arguments deviation
- 36 Downloads
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.
Keywords and phrasesFisher–Kolmogorov equation diffusion spatial deviation delay differential equation normal form numerical analysis
Unable to display preview. Download preview PDF.
- 1.A. N. Kolmogorov, I. G. Petrovsky, and N. S. Piskunov, Bull. Univ. d’EtatMoscou, Ser. A 1, 1–26 (1937).Google Scholar
- 5.S. Kakutani and L. Markus, Ann. Math. Stud. 4, 1–18 (1958).Google Scholar
- 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.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
- 11.S. D. Glyzin, Model. Anal. Inform. Sist. 16 (3), 96–116 (2009).Google Scholar