Existence and stability of non-monotone travelling wave solutions for the diffusive Lotka–Volterra system of three competing species

Abstract

This paper considers the problem: if coexistence occurs in the long run when a third species w invades an ecosystem consisting of two species u and v on , where u, v and w compete with one another. Under the assumption that the influence of w on u and v is small and other suitable conditions, we show that the three species can coexist as a non-monotone travelling wave. Such type of non-monotone waves plays an important role in the study of three-species phenomena. However, fewer results are known for the existence of such waves in the literature. Our approach, based on the method of sub- and super-solutions and bifurcation theory, provides a new approach to construct non-monotone waves of this type. Moreover, we show that the waves we construct are stable. To the best of our knowledge, this is the first rigorous result of stability for such type of waves.