On the ratio of total masses of species to resources for a logistic equation with Dirichlet boundary condition

TL;DR

This paper investigates the maximum ratio of total species mass to resources in a diffusive logistic equation with Dirichlet boundary conditions, extending previous results from Neumann problems to Dirichlet cases.

Contribution

It demonstrates that the supremum of the ratio is 3 in one dimension and infinity in higher dimensions for the Dirichlet problem, similar to Neumann cases.

Findings

Supremum of ratio is 3 in 1D

Supremum is infinite in multi-dimensional balls

Results extend previous Neumann problem findings

Abstract

We consider the stationary problem for a diffusive logistic equation with the homogeneous Dirichlet boundary condition. Concerning the corresponding Neumann problem, Wei-Ming Ni proposed a question as follows: Maximizing the ratio of the total masses of species to resources. For this question, Bai, He and Li showed that the supremum of the ratio is 3 in the one dimensional case, and the author and Kuto showed that the supremum is infinity in the multi-dimensional ball. In this paper, we show the same results still hold true for the Dirichlet problem. Our proof is based on the sub-super solution method and needs more delicate calculation because of the range of the diffusion rate for the existence of the solution.