Asymptotic solutions of the 1D nonlocal Fisher-KPP equation

Two analytical methods have been developed for constructing approximate solutions to a nonlocal generalization of the 1D Fisher– Kolmogorov–Petrovskii–Piskunov equation. This equation is of special interest in studying the pattern formation in microbiological populations. In the greater part of the paper, we consider in detail a semiclassical approximation method based on the WKB–Maslov theory under the supposition of weak diffusion. The semiclassical asymptotics are sought in a class of trajectory concentrated functions. Such functions are localized in a neighborhood of a point moving in space. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval which can be small in the sense that a pattern has no time to form in this interval. In the final part of the paper, we have constructed asymptotics which are different from the semiclassical ones and can describe the evolution of the solutions of the Fisher–Kolmogorov–Petrovskii–Piskunov equation at large times. These asymptotics represent small perturbations on the background of Theoretical Physics Department, National Research Tomsk State University, 36 Lenin ave., 634050 Tomsk, Russia Laboratory of Mathematical Physics of Mathematical Physics Department, National Research Tomsk Polytechnic University, 30 Lenin ave., 634050 Tomsk, Russia