Spatiotemporal Dynamics of a Mussel-Algae Model on the Square Domain

We investigate the spatiotemporal dynamics of a non-local mussel-algae model, defined on a square domain with time delays and Neumann boundary conditions. Initially, we examine the well-posedness of the solutions. By analyzing the multiplicity of eigenvalues, we establish the existence of both Hopf and equivariant Hopf bifurcations. Using tools such as phase space decomposition, center manifold reduction, equivariant Hopf bifurcation theory, and the normal form method, we derive third-order truncated normal forms near the equivariant Hopf bifurcation point. This allows us to classify the system's spatiotemporal patterns into ten distinct types within the parameter plane. Unlike models constructed on one-dimensional domains, the two-dimensional symmetric model demonstrates more complex dynamic behaviors, including standing waves, rotating waves, stripes, and spots. Numerical simulations not only corroborate the theoretical predictions but also align with field observation in ecological systems, shedding light on the mechanisms underlying the formation of regular patterns due to the behavioral aggregation of mussels.