A logistic equation with nonlocal interactions

TL;DR

This paper analyzes a nonlocal logistic equation modeling populations with nonlocal diffusion and proliferation, comparing local and nonlocal cases, and providing explicit survival thresholds across various environments.

Contribution

It introduces explicit thresholds for population survival in nonlocal logistic models and compares their behavior with local models across different spatial settings.

Findings

Nonlocal populations better adapt to sparse resources.

Explicit survival thresholds depend on spectral properties of the domain.

Nonlocal models can outperform local ones in small or sparse environments.

Abstract

We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms. More precisely, for populations that propagate according to a L\'evy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case. As ambient space, we can consider: bounded domains, periodic environments, and transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one. In each of these cases, we analyze the existence/nonexistence of solutions in terms of the spectral properties of the domain. In particular, we give a detailed description of the fact that nonlocal populations may better adapt to sparse resources and small environments.