Forced waves in a Lotka-Volterra competition-diffusion model with a shifting habitat

We establish the existence of traveling waves for a Lotka-Volterra competition-diffusion model with a shifting habitat. It is assumed that the growth rate of each species is nondecreasing along the x-axis, positive near ∞ and negative near −∞, and shifting rightward at a speed c. We show that under appropriate conditions, for the case that one species is competitively stronger near ∞ and the case that both species coexist near ∞, there exists a critical number c ¯ ( ∞ ) such that for any c > c ¯ ( ∞ ) there exists a forced traveling wave with speed c connecting the origin and a semi-trivial steady state and for c c ¯ ( ∞ ) such a traveling wave does not exist. We also show that when a coexistence steady state exists, for any c > 0 , there is a forced traveling wave with speed c connecting the origin and the coexistence steady state.