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Wavefront solutions of quasilinear reaction–diffusion systems with mixed quasi-monotonicity

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 for . Each function is quasi-monotone increasing for some components of and decreasing for other components of . 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.