Effects of nonlocal dispersal strategies and heterogeneous environment on total population

TL;DR

This paper analyzes how nonlocal dispersal strategies and environmental heterogeneity influence total population, revealing bounds and optimal conditions that differ from local dispersal models.

Contribution

It establishes an upper bound for population-resource ratio and identifies conditions for a unique maximum population with respect to dispersal rate in nonlocal models.

Findings

Upper bound for population-resource ratio is proportional to sqrt of dispersal rate

Constructed examples show the bound is optimal

Identified conditions for a single population maximum as dispersal rate varies

Abstract

In this paper, we consider the following single species model with nonlocal dispersal strategy $$ d\mathcal {L} [\theta] (x,t) + \theta(x,t) [m(x)- \theta(x,t)]=0 , $$ where $\mathcal {L}$ denotes the nonlocal diffusion operator, and investigate how the dispersal rate of the species and the distribution of resources affect the total population. First, we show that the upper bound for the ratio between total population and total resource is $C\sqrt{d}$. Moreover, examples are constructed to indicate that this upper bound is optimal. Secondly, for a type of simplified nonlocal diffusion operator, we prove that if $ \frac{\sup m}{\inf m}<\frac{\sqrt{5}+1}{2}$, the total population as a function of dispersal rate $d$ admits exactly one local maximum point in $\displaystyle (\inf m, \sup m)$. These results reveal essential discrepancies between local and nonlocal dispersal strategies.