Nonlinear stability of forced traveling waves for a Lotka–Volterra cooperative model under climate change

Abstract

This paper is concerned with the nonlinear stability of forced traveling waves for a Lotka–Volterra cooperative model under climate change. Firstly, by applying the ‐weighted energy estimate method, the comparison principle, and the squeezing technique, we investigate that all forced traveling waves with the speed are exponentially stable in the form of for some . Secondly, in order to improve the former results, we take another weight function and construct the related different weighted Sobolev space. Instead of the ‐weighted energy estimate, we first establish a ‐weighted energy estimate in the weighted Sobolev space. Then, by using this crucial ‐estimate, we further obtain the desired ‐energy estimate. Finally, we obtain that all forced traveling waves with the speed are exponentially stable in the form of for some , where .