Optimisation of total population in logistic model with nonlocal dispersals and heterogeneous environments

TL;DR

This paper explores how to maximize the total population in a logistic model with nonlocal dispersal and heterogeneous environments, revealing key differences from local dispersal models and providing bounds related to diffusion rates.

Contribution

It establishes bounds on the maximum total population in nonlocal dispersal models and compares these with local diffusion models, highlighting significant differences.

Findings

For diffusion rates d ≥ 1, total population bounds scale with √d.

In one-dimensional cases with random diffusion, the maximum population equals 3 times the L^1 norm of m.

Discrepancies are shown between local and nonlocal dispersal strategies.

Abstract

In this paper, we investigate the issue of maximizing the total equilibrium population with respect to resources distribution m(x) and diffusion rates d under the prescribed total amount of resources in a logistic model with nonlocal dispersals. Among other things, we show that for $d\ge1$, there exist $C_0, C_1>0$, depending on the $\|m\|_{L^1}$ only, such that $$C_0\sqrt{d}<\mbox{supremum~ of~ total~ population}<C_1\sqrt{d}.$$ However, when replaced by random diffusion, a conjecture, proposed by Ni and justified in [3], indicates that in the one-dimensional case, supremum of total population$=3\|m\|_{L^1}$. This reflects serious discrepancies between models with local and nonlocal dispersal strategies.