Impact of Nonlinear Diffusion and Heterogeneity on Competing Populations Dynamics

The impact of migration effects on the formation of population distributions is studied. We consider the model of interplay between two species (resident and invader), and apply the theory of cosymmetry to classify different population scenarios. The system of reaction – diffusion – advection equations is used to describe the nonlinear diffusion and taxis because of nonuniform distribution of the resource. The logistic law of growth is taken to model local interaction between species. We consider a one-dimensional habitat with no-flux boundary conditions. Finite-difference discretization with a staggered grid is used for the spatial coordinate and Runge – Kutta integrator to solve the resulting system of ordinary differential equations of large order. A computer experiment is applied to analyze the dynamics of populations and migration fluxes. We numerically build the maps of migration parameters for description scenarios of invasion and competition. It is found that nonlinear diffusion has an influence on invasion because intraspecific taxis compensates nonoptimal migration to resource. Negative coefficients of intraspecific taxis stimulate diffusion for both species and prevent excessive concentration of populations. This aids the coexistence of species as stationary distributions. Different coefficient signs imply the implementation of corresponding stable semipositive solutions. Direct numerical experiments show that the coexistence of species occurs at large positive coefficients of intraspecific taxis. The dependence of the scenario on the initial distributions is established.