Wave propagation in the Kolmogorov–Petrovskii–Piskunov problem with delay

Abstract

The problem of density wave propagation governed by a logistic equation with delay and diffusion (Fisher–Kolmogorov–Petrovskii–Piskunov equation with delay) was studied. To analyze the qualitative behavior of solutions to this equation with periodic boundary conditions in the case of the diffusion parameter tending to zero, the normal form of the problem, i.e., the Ginzburg–Landau equation was constructed near the unit equilibrium. A numerical analysis of wave propagation showed that, for sufficiently small delays, this equation has solutions close to those of the standard Kolmogorov–Petrovskii–Piskunov equation. As the delay parameter increases, a decaying oscillatory component appears in the spatial distribution of the solution and, then, undamped (in time) and slowly propagating (in space) oscillations close to solutions of the corresponding boundary value problem with periodic boundary conditions are observed near the initial perturbation segment.