Mathematical model of the ideal distribution of related species in a non-hogeneous environment

We consider a system of nonlinear equations of parabolic type, which models the dynamics of competing species in a heterogeneous environment with directed migration and the dependence of parameters on spatial variables. We found relations for the diffusion and migration coefficients of the system, under which the initial-boundary value problem has explicit solutions, united in a continuous family of stationary distributions of populations. These solutions (equilibria) correspond to ideal free distributions and correspond to cosymmetry in the problem's subspace. Using the theory of V. I. Yudovich concerning the destruction of cosymmetry, the perturbation of equations for a system of two related types was analytically studied. The case of disappearing of the family of equilibria is studied when only isolated solutions remain. We derived the conditions on the parameters when an additional equilibrium arises corresponding to the coexistence of species. For a system of two related populations competing for a common resource in a one-dimensional area, finite-difference approximations are constructed based on a staggered grid scheme. Calculations show the non-identity of the stability spectrum of stationary distributions from the family.