Stability of non-critical waves to a cooperation-competition model with three species

In this paper, we prove the stability of traveling wave solutions for a three-species cooperation-competition system. To ensure the existence of traveling wave fronts, i.e. monotone traveling wave solutions, for this system, the wave speed $c$ must be greater than or equal to a critical value $c_{\ast }$. In the existing literature, the weighted energy method has been employed to prove that traveling wave fronts are stable when $c$ is sufficiently larger than $c_{\ast }$. This paper improves this result by showing that, under certain conditions on the system parameters, traveling wave fronts with $c>c_{\ast }$ (i.e., non-critical waves) are stable. First, by virtue of the asymptotic behavior and positivity of traveling wave fronts, we establish some estimates. Then, we prove the estimates for solutions to the perturbed system. Finally, the exponential stability of non-critical waves is obtained by the weighted energy method and the squeezing technique. Our conclusions expand the range of speeds for which traveling wave fronts are stable, thereby strengthening existing conclusions.