ИСТИНА

In this work we consider some initial-boundary value problem for a singular perturbed reaction-diffusion system. Such systems are often applied for modeling front propagation in physical, chemical or biological processes. This particular system arises in solving of some urban ecosystem model [1]. We study this problem with asymptotic and numerical methods. The problem is stiff, so it causes asymptotic analysis that is able to help us to extract some \emph{a priory} information about solution (e.g. about form and localization of the moving fronts) in order to significantly improve further numerical calculations. With asymptotic methods we also define the conditions for existence of the traveling-wave solution [2]. We propose new algorithm for solving of this problem that based on implementation of 1) method of lines that reduces the initial system of partial differential equations to the stiff system of ordinary differential equations, 2) very stable Rosenbrock scheme with complex coefficient for stiff system of ordinary differential equations [3], 3) the Richardson extrapolation, 4) a dynamic adaptive mesh which construction based on \emph{a priory} information that has been extracted by asymptotic analysis.