Forced waves and gap formations for a three-species competitive-cooperative system with nonlocal dispersal and shifting habitats

This paper is concerned with the existence of forced waves and gap formations for a three-species competitive-cooperative system with nonlocal dispersal under habitat shifting at a constant speed, in which two species cooperate with each other while simultaneously competing with the third species together. First, the existence of forced waves is established by means of iterative techniques, where the speed lies within a specified interval. In the existence part, kernel functions are asymmetric, and the growth functions are allowed to be sign-preserving. Then, under the conditions that the kernel functions are symmetric and that the growth functions change sign, the gap formations are derived via a comparison principle.