Limiting Structure of Shrinking Solutions to the Stationary Shigesada--Kawasaki--Teramoto Model with Large Cross-Diffusion

Abstract

This paper is concerned with the limiting behavior of coexistence steady states of the Lotka--Volterra competition model as a cross-diffusion term tends to infinity. Under the Neumann boundary condition, Lou and Ni [J. Differential Equations, 154 (1999), pp. 157--190] derived a couple of limiting systems, which characterize the limiting behavior of coexistence steady states. One of two limiting systems characterizing the segregation of the competing species has been studied by Lou, Ni, and Yotsutani [Discrete Contin. Dyn. Syst., 10 (2004), pp. 435--458], and their work revealed the detailed bifurcation structure for the one-dimensional (1D) case. This paper focuses on the other limiting system characterizing the shrinkage of the species which is not endowed with the cross-diffusion effect. The bifurcation structure of positive solutions to the limiting system is stated. In particular, for the 1D case, we obtain a global connected set of solutions that bifurcates from a point on the line of constant solutions and blows up where the birth rate of the species is equal to the least positive eigenvalue of $-\Delta$ subject to the homogeneous Neumann boundary condition.