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 $$\begin{array}{c}\hfill \mathit{\partial }{u}_i/\mathit{\partial }t=\mathrm{\nabla }·\left(D_i\left(u_i\right)\mathrm{\nabla }{u}_i\right)+f_i\left(\mathbf{u}\right)-\infty <x<\infty ,t>0\end{array}$$

for $i=1,\dots ,n$. Each function $f_i$ is quasi-monotone increasing for some components of $\mathbf{u}=\left(u_1,\dots ,u_n\right)$ and decreasing for other components of $\mathbf{u}$. 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.

KEYWORDS:

• Quasilinear reaction–diffusion systems
• mixed quasi-monotonicity
• existence theorem for wavefront solutions
• upper–lower solution method
• coexistence in competition models

AMS SUBJECT CLASSIFICATIONS:

• 35C07
• 35K57
• 35K59
• 92D25

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments that helped in strengthening the results in this article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially support by a grant from the Simons Foundation [grant number 245488 to Weihua Ruan].