Wave propagation in the Kolmogorov–Petrovsky–Piscounov–Fisher equation with delay

Abstract

The problem of density wave propagation is considered for a logistic equation with delay and diffusion. This equation, called the Kolmogorov–Petrovsky–Piscounov–Fisher equation with delay, is investigated by asymptotic and numerical methods. Local properties of solutions corresponding to this equation with periodic boundary conditions are studied. It is shown that an increase in the period leads to the emergence of stable solutions with a more complex spatial structure. The process of wave propagation from one and from two initial perturbations is analyzed numerically, which allows tracing the process of wave interaction in the second case. The complex spatially inhomogeneous structure arising during the wave propagation and interaction can be explained by the properties of the corresponding solutions of a periodic boundary value problem with an increasing range of the spatial variable.