Limiting profile for stationary solutions maximizing the total population of a diffusive logistic equation

Abstract

This paper focuses on the stationary problem of the diffusive logistic equation on a bounded interval. We consider the ratio of a population size of a species to a carrying capacity which denotes a spatial heterogeneity of an environment. In one-dimensional case, Wei-Ming Ni proposed a variational conjecture that the supremum of this ratio varying a diffusion coefficient and a carrying function is 3. Recently, Xueli Bai, Xiaoqing He, and Fang Li [Proc. Amer. Math. Soc. 144 (2016), pp. 2161–2170] settled the conjecture by finding a special sequence of diffusion coefficients and carrying functions. Our interest is to derive a profile of the solutions corresponding to this maximizing sequence. Among other things, we obtain the exact order of the maximum and the minimum of solutions of the sequence. The proof is based on separating the stationary problem into two ordinary differential equations and smoothly adjoining each solution.