Existence of traveling wave solutions for density-dependent diffusion competitive systems

In this paper we are concerned with the existence of traveling wave solutions for two species competitive systems with density-dependent diffusion. Since the density-dependent diffusion is a kind of nonlinear diffusion and degenerates at the origin, the methods for proving the existence of traveling wave solutions for competitive systems with linear diffusion are invalid. To overcome the degeneracy of diffusion, we construct a nonlinear invariant region Ω near the origin. Then by using the method of phase plane analysis, we prove the existence of traveling wave solutions connecting the origin and the unique coexistence state, when the speed c is large than some positive value. In addition, when one species is density-dependent diffusive while the other is linear diffusive, via the change of variables and the central manifold theorem, we prove the existence of the minimal speed c∗ . And for c⩾c∗ , traveling wave solutions connecting the origin and the unique coexistence state still exist. In particular, when c=c∗ , we find that one component of the traveling wave solution is sharp type while the other component is smooth, which is a different phenomenon from linear diffusive systems and scalar equations.