An optimization problem and its application in population dynamics

Bai, Xueli; He, Xiaoqing; Li, Fang

doi:10.1090/proc/12873

An optimization problem and its application in population dynamics
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by Xueli Bai, Xiaoqing He and Fang Li PDF

Proc. Amer. Math. Soc. 144 (2016), 2161-2170 Request permission

Abstract:

This paper is concerned with a diffusive logistic model in population ecology. As observed by Y. Lou, in a spatially heterogeneous environment, this model can always support a total population at equilibrium greater than the total carrying capacity. In other words, the ratio of the total population at equilibrium to the total carrying capacity is always larger than $1$. Our goal is to find the supremum of this ratio taken over all possible choices of spatial distributions of resources and the species’ dispersal rate. A conjecture proposed by W.-M. Ni is that, in the one-dimensional case, the supremum is $3$. We settle this conjecture and then apply our result to study the global dynamics of a heterogeneous Lotka-Volterra competition-diffusion system.

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