The nonlocal dispersal equation with seasonal succession

TL;DR

This paper studies a nonlocal dispersal equation with seasonal changes, establishing conditions for the existence, uniqueness, and long-term behavior of solutions based on the principal eigenvalue.

Contribution

It proves the existence and uniqueness of global positive solutions and characterizes their long-term dynamics depending on the principal eigenvalue.

Findings

Solutions tend to zero when the principal eigenvalue is non-negative.

A unique globally stable positive solution exists when the principal eigenvalue is negative.

The dynamics are fully determined by the sign of the principal eigenvalue.

Abstract

In this paper, we focus on the nonlocal dispersal monostable equation with seasonal succession, which can be used to describe the dynamics of species in an environment alternating between bad and good seasons. We first prove the existence and uniqueness of global positive solution, and then discuss the long time behaviors of solution. It is shown that its dynamics is completely determined by the sign of the principal eigenvalue, i.e., the time periodic problem has no positive solution and the solution of the initial value problem tends to zero when principal eigenvalue is non-negative, while the time periodic positive solution exists uniquely and is globally asymptotically stable when principal eigenvalue is negative.