Abstract
We propose an approximate analytical approach to a ( 1 + 1 ) dimensional two-component system consisting of a nonlocal generalization of the well-known Fisher–Kolmogorov–Petrovskii– Piskunov (KPP) population equation and a diffusion equation for the density of the active substance solution surrounding the population. Both equations of the system have terms that describe the interaction effects between the population and the active substance. The first order perturbation theory is applied to the system assuming that the interaction parameter is small. The Wentzel–Kramers–Brillouin (WKB)–Maslov semiclassical approximation is applied to the generalized nonlocal Fisher–KPP equation with the diffusion parameter assumed to be small, which corresponds to population dynamics under certain conditions. In the framework of the approach proposed, we consider symmetry operators which can be used to construct families of special approximate solutions to the system of model equations, and the procedure for constructing the solutions is illustrated by an example. The approximate solutions are discussed in the context of the released activity effect variously debated in the literature.
Highlights
Investigations of non-local models of reaction–diffusion (RD) systems with long-range interactions are a developing trend in modern nonlinear physics and mathematics ranging from condensed matter physics to physics of living systems.RD systems are in the focus of researchers due to the spatial and temporal patterns that can be formed in such systems in the process of their evolution under certain conditions
RD population model which describes a population interacting with an active substance
We apply the first-order perturbation method to the model equations in the interaction parameter and obtain that finding the leading term of the perturbation solution is reduced to solving the non-local generalized Fisher–Kolmogorov–Petrovskii– Piskunov (KPP) equation
Summary
Investigations of non-local models of reaction–diffusion (RD) systems with long-range interactions are a developing trend in modern nonlinear physics and mathematics ranging from condensed matter physics to physics of living systems. The approach uses the perturbation method with a small parameter to describe the interaction between population density and active substance density and the WKB–Maslov method of semiclassical asymptotics assuming a weak diffusion for solving the non-local Fisher–KPP type equation with an interaction term [23]. We would like to note that the approximate analytical approach offered here for RD population dynamical systems with long-range interactions can have wider applications in solving various problems of nonlinear physics ranging from quantum matter wave models to cosmology (see, e.g., [30,31] and references therein).
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