Dynamics Analysis of a Discrete Mussel–Algae System with Diffusion and Advection Considering the Half-Saturation Constant

The spatiotemporal dynamics of a discrete mussel–algae system with diffusion and advection considering the half-saturation constant are discussed in this paper. A new perspective for the study on the relationship between the population densities of mussels and algae is presented by the discrete mussel–algae interaction model. By choosing the time step length and the advection coefficient as bifurcation parameters, several dynamical behaviors of the model are studied. The conditions for local asymptotic stability, Flip bifurcation and Neimark–Sacker bifurcation in the absence of diffusion are obtained. By using the center manifold theorem and bifurcation theory, the normal forms and direction of these two bifurcations are derived. Considering the system with diffusion and advection, the critical scenarios for spatial instability, including pure Turing instability, Flip–Turing instability and Neimark–Sacker–Turing instability, are established. Numerical simulations are given to verify the proposed results and illustrate the complicated dynamical behaviors of the mussel–algae system. Multiple spatial patterns are displayed, and these patterns tend to be distributed along the horizontal direction due to the presence of an advection term. Supported by numerical simulations, we conclude that not only self-diffusion can make the previously stable system become spatial Turing instability, but also the advection term (i.e. tidal flow) can change the spatiotemporal stability of the system, resulting in richer dynamical phenomena. The discretization perspective and the theoretical results regarding pattern formation of certain discrete models with advection might give rise to significant results for further research.