A qualitative analysis of positive steady-state solutions for a mussel-algae model with diffusion

In this paper, a diffusive mussel-algae model subject to Neumann boundary conditions is considered. The main criteria for the stability and instability of the constant steady-state solutions are presented. Then, by the maximum principle, Hölder inequality and Poincar$ \acute{e} $ inequality, a priori estimates and some characters of positive solutions are given and the nonexistence of the non-constant steady-state solutions for the corresponding elliptic system is investigated. Moreover, the steady-state bifurcations at both simple and double eigenvalues are intensively investigated. In particular, the implicit function theorem and the techniques of space decomposition are used to get the local structure of steady-state bifurcations from double eigenvalues. Next, our analysis focuses on providing specific conditions that can determine the local bifurcation direction and extend the local bifurcation to the global one. Finally, the numerical results are presented to provide support and complement the theoretical analysis findings. More specifically, under various parameters, the evolution processes in spatial patterns are illustrated.