Traveling wave solutions of Lotka–Volterra competition systems with nonlocal dispersal in periodic habitats

Abstract

This paper is concerned with space periodic traveling wave solutions of the following Lotka–Volterra competition system with nonlocal dispersal and space periodic dependence,{∂u1∂t=∫RNκ(y−x)u1(t,y)dy−u1(t,x)+u1(a1(x)−b1(x)u1−c1(x)u1),x∈RN∂u2∂t=∫RNκ(y−x)u2(t,y)dy−u2(t,x)+u2(a2(x)−b2(x)u1−c2(x)u2),x∈RN. Under suitable assumptions, the system admits two semitrivial space periodic equilibria (u1⁎(x),0) and (0,u2⁎(x)), where (u1⁎(x),0) is linearly and globally stable and (0,u2⁎(x)) is linearly unstable with respect to space periodic perturbations. By sub- and supersolution techniques and comparison principals, we show that, for any given ξ∈SN−1, there exists a continuous periodic traveling wave solution of the form (u1(t,x),u2(t,x))=(Φ1(x−ctξ,ctξ),Φ2(x−ctξ,ctξ)) connecting (u1⁎(⋅),0) and (0,u2⁎(⋅)) and propagating in the direction of ξ with speed c>c⁎(ξ), where c⁎(ξ) is the spreading speed of the system in the direction of ξ. Moreover, for c<c⁎(ξ) there is no such solution. When the wave speed c>c⁎(ξ), we also prove the asymptotic stability and uniqueness of traveling wave solution using squeezing techniques.