Coexistence States for Some Population Models with Nonlinear Cross-Diffusion

Yoshio Yamada

Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku, Tokyo 169, Japan

"Dedicated to Professor Kyûya Masuda on the occasion of his sixtieth birthday"

(Received April 10, 1996; Accepted June 28, 1996)

Keywords: Lotka-Volterra Population Model, Nonlinear Cross-Diffusion, Coexistence State

Abstract. We consider Lotka-Volterra population models with nonlinear diffusion of the form

u_t = m{(1 + av)u} + f(u, v),  v_t = n{(1 + bu)v} + g(u, v)

where f and g represent reaction terms of competition type or prey-predator type. Addition of diffusion terms to ordinary differential equations (ODEs) yields new aspects which are not predicted from the analysis of ODEs alone. The present paper is mainly concerned with positive steady-states (coexistence states) for the above system with zero Neumann or Dirichlet boundary conditions. From the view-point of mathematics as well as population biology, we will show some important features which are induced by the combination of nonlinear cross-diffusion with reaction kinetics.