Wavefront solutions of quasilinear reaction–diffusion systems with mixed quasi-monotonicity. Issue 5 (4th April 2019) - Ruan, W. H.; Feng, Wei; Lu, Xin
ABSTRACT: We study the existence of wavefront solutions in a general class of mixed quasi-monotone reaction–diffusion systems with nonlinear diffusions in the form∂ u i / ∂ t = ∇ · D i u i ∇ u i + f i u - ∞ < x < ∞, t > 0 fori = 1, …, n . Each functionf i is quasi-monotone increasing for some components ofu = u 1, …, u n and decreasing for other components ofu . Such systems model reaction–diffusion processes with a density driven diffusion mechanism. Under certain general conditions we prove the existence of a traveling wave solution that is between a pair of coupled upper and lower solutions. As examples, we discuss a predator–prey model and a multi-species competition model, both with nonlinear diffusions. In both models, the presences of wavefront solutions flowing towards the coexistence states are established as illustrations for applications of our main result, with numerical simulations.