On logistic models with a carrying capacity dependent diffusion: Stability of equilibria and coexistence with a regularly diffusing population

We consider the reaction–diffusion equation describing the population with the logistic type of growth and diffusion stipulated by the carrying capacity KK, which leads to the term DΔ(u/K)DΔ(u/K), where uu is the population level. In the logistic model the introduction of the standard diffusion term ΔuΔu (incorporated with the zero Neumann boundary conditions) leads to the situation when the population tends to be equally distributed over the space available, even if the carrying capacity K(x)K(x) varies significantly with location. The strategy with a KK-driven diffusion is compared to the model with standard diffusion, and we demonstrate that for two competing populations with two different strategies, the equilibrium where only the species which follows KK-driven diffusion survives, is globally asymptotically stable.