Persistence and Spread of Solutions in a Two-Species Lotka--Volterra Competition-Diffusion Model with a Shifting Habitat

We consider a two-species Lotka--Volterra competition-diffusion model with a shifting habitat. The growth rate of each species is nondecreasing along the x-axis, and it changes sign and shifts rightward at a speed c. We investigate the population dynamics of the model in the habitat suitable for growth of both species for two cases: (i) one species is competitively stronger and has a slower spreading speed, and (ii) both species coexist. We obtain conditions under which the outcome of competition depends critically on a number c̄(∞) given by the model parameters. We show that under appropriate conditions, if c̄(∞)>c, then the species with the faster spreading speed spreads into the open space at its own speed and the species with the slower spreading speed spreads into its rival at speed c̄(∞), and if $c̄(∞)