Dynamic properties of the Fisher–Kolmogorov–Petrovskii–Piscounov equation with the deviation of the spatial variable

We consider the problem of the density wave propagation of a logistic equation with the deviation of the spatial variable and diffusion (the Fisher–Kolmogorov equation with the deviation of the spatial variable). The Ginzburg–Landau equation was constructed in order to study the qualitative behavior of the solution near the equilibrium state. We analyzed the profile of the wave equation and found conditions for the appearance of oscillatory regimes. The numerical analysis of the wave propagation shows that, for a fairly small spatial deviation, this equation has a solution similar to that the classical Fisher–Kolmogorov equation. An increase in this spatial deviation leads to the existence of the oscillatory component in the spatial distribution of solutions. A further increase in the spatial deviation leads to the destruction of the traveling wave. This is expressed in the fact that undamped spatiotemporal fluctuations exist in a neighborhood of the initial perturbation. These fluctuations are close to the solution of the corresponding boundary value problem with periodic boundary conditions. Finally, when the spatial deviation is large enough we observe intensive spatiotemporal fluctuations in the whole area of wave propagation.