Existence of Coexistence States for a Strongly Coupled Cooperative Population Model With Cross‐Diffusion

ABSTRACT

This article investigates the existence of coexistence states for a strongly coupled cooperative elliptic system with cross‐diffusion, modeling two mutualistic species. By employing the method of upper and lower solutions coupled with monotone iterative techniques, we establish sharp sufficient conditions for the existence of positive coexistence states, revealing that strong interspecific cooperation and sufficiently large intrinsic growth rates can overcome the competitive exclusion induced by cross‐diffusion in hostile environments. A novel transformation is introduced to handle the strongly coupled structure, and numerical simulations are provided to validate the main theoretical findings.