Mathematical model of evolutionary stable strategy for population under conditions of competition in a heterogeneous habitat

Based on the diffusion-reaction equations, a system of two related populations competing for a single resource is considered. A simple mathematical formulation of the problem under study is given, taking into account diffusion processes and local interactions between members of the stud-ied biological community in conditions of a heterogeneous habitat. Using the concept of ideal free distribution (IFD), a model of an evolutionarily stable strategy (ESS) is constructed for one of the two populations, leading to its long-term sustainable existence. Stationary solutions of the system and the conditions for their stability in a one-dimensional area with an uneven distribution of the resource are analyzed. The principles (rules of conduct) have been formulated, the achievement of which ensures the implementation of the ESS of the selected population. A modified integro-interpolation method with discretization based on shifted grids is used for the numerical solution of the initial-boundary value problem. Based on the SPECIES-21 software package implemented in the MATLAB environment, a series of computational experiments was conducted to establish the degree of influence of three key factors (initial population distributions, diffusion coefficients, and competitive interaction) on the behavior of the system.