Traveling waves of a reaction-diffusion system with partially coupled diffusion

Abstract

This work is devoted to the study of traveling wave solutions of reaction-diffusion systems, where the diffusion rate of~$v$, the second component, depends on~$u$, the first component. Such systems arise in prey-predator models, where the predator~$v$ is actively hunting its prey~$u$, or in epidemiological models where the disease induces erratic behavior, for example rabies. This results in a quasilinear coupling in the highest-order term of the equation for~$v$. From the theoretical point of view, we fully classify traveling wave solutions when the first component does not diffuse, in which case the problem can be reduced to a nonlinear scalar equation. The case where both components diffuse is investigated numerically.