Asymptotics of solutions of finite-difference approximations of a logistic equation with delay and small diffusion

Abstract

We studied the dynamics of finite-difference approximation with respect to spatial variables of a logistic equation with delay and diffusion. It was assumed that the diffusion coefficient is small and the Malthusian coefficient is large. The question concerning the existence and asymptotic behavior of the attractors was studied using special asymptotic methods. It has been shown that there is a rich set of attractors of different types in the phase space: guiding centers, systems of helicon waves, etc. The major asymptotic characteristics of all the solutions from the corresponding attractors are adduced in this work. Typical graphics of the motion of wave fronts of different structures are represented in the paper.