Wave propagation in the Kolmogorov–Petrovskii–Piskunov–Fischer equation with time delay and spatial deviation

Abstract

For the logistic equation with diffusion two generalizations are considered: one with time delay and the other with deviation in the spatial variable. For these equations, the problem of density wave propagation is considered. These equations, known as Kolmogorov–Petrovsky–Piskunov–Fisher equations with delay or deviation (KPPF with delay or deviation), are investigated using both asymptotic and numerical methods.To investigate the qualitative behavior of solutions of such equations, the wave profile equation is considered and the conditions for the emergence of its oscillatory modes are identified.The local properties of solutions with periodic boundary conditions are examines, revealing that an increase in the period leads to the emergence of stable solutions with a more complex spatial structures.Numerical analysis of the wave propagation process indicates that at sufficiently small delay values, this equation has solutions that are close to those of the standard KPPF equation. Increasing the delay parameter leads first to the appearance of a damped oscillatory component in the spatial distribution of the solution. With further increases of this parameter, a complex spatial profile of the wave appears.In the final part of the chapter, which focuses on numerical calculation, the numerical analysis of the wave propagation process from two initial perturbations is performed. This analysis tracks the process of wave interaction. The complex spatially inhomogeneous structure that arises during wave propagation and interaction can be explained by the properties of the corresponding solutions of the periodic boundary value problem, especially as the interval of change in the spatial variable increases.