Propagation Phenomena for a Nonlocal Dispersal Lotka–Volterra Competition Model in Shifting Habitats

Abstract

This paper is concerned with the propagation phenomena for a nonlocal dispersal Lotka–Volterra competition model with shifting habitats. It is assumed that the growth rate of each species is nondecreasing along the x-axis, positive near $\infty$ and nonpositive near $-\infty$, and shifting rightward with a speed $c>0$. In the case where both species coexist near $\infty$, we established three types of forced waves connecting the origin, respectively to the coexistence state with any forced speed c; to itself with forced speed $c>c^{*}(\infty )$; and to a semi-trivial steady state with forced speed $c>{\bar{c}}(\infty )$, where $c^{*}(\infty )$ and ${\bar{c}}(\infty )$ are two positive numbers. In the case where one species is competitively stronger near $\infty$, we also obtain the existence and nonexistence of forced waves connecting the origin to the semi-trivial steady state. Our results show the existence of multiple types of forced waves with the same forced speed. The mathematical proofs involve integral equations and Schauder’s fixed point theorem, and heavily rely on the construction of various upper-lower solutions, which adds new techniques to deal with the “shifting environments” problem.