A study of nonlocal spatially heterogeneous logistic equation with harvesting

TL;DR

This paper investigates nonlocal reaction-diffusion equations with harvesting, analyzing existence, uniqueness, and long-term behavior of solutions using analytic and probabilistic methods, covering various fractional operators.

Contribution

It introduces a comprehensive study of steady states and dynamics for a broad class of nonlocal operators including fractional Laplacians and relativistic operators.

Findings

Existence and uniqueness of steady state solutions.

Conditions for multiple solutions.

Asymptotic behavior of solutions over time.

Abstract

We study a class of nonlocal reaction-diffusion equations with a harvesting term where the nonlocal operator is given by a Bernstein function of the Laplacian. In particular, it includes the fractional Laplacian, fractional relativistic operators, sum of fractional Laplacians of different order etc. We study existence, uniqueness and multiplicity results of the solutions to the steady state equation. We also consider the parabolic counterpart and establish the long time asymptotic of the solutions. Our proof techniques rely on both analytic and probabilistic arguments.