Chapter 6 Positive solutions for Lotka-Volterra systems with cross-diffusion

This article is concerned with reaction-diffusion systems with nonlinear diffusion effects, which describe competition models and prey-predator models of Lotka-Volterra type in population biology. The system consists of two nonlinear diffusion equations where two unknown functions denote population densities of two interacting species. The main purpose is to discuss the existence and nonexistence of positive steady state solutions to such systems. Here a positive solution corresponds to a coexistence state in population models. We will derive a priori estimates of positive solutions by maximum principle for elliptic equations and employ the degree theory on a positive cone to show the existence of a positive solution. The existence results can be reconsidered from the view-point of bifurcation theory. We will give some information on the direction of bifurcation of positive solutions and their stability properties in terms of some biological coefficients. Moreover, we will also study the existence of multiple positive solutions for a certain class of prey-predator systems with nonlinear diffusion by making one of cross-diffusion coefficients sufficiently large.