Spatiotemporal patterns near the Turing-Hopf bifurcation in a delay-diffusion mussel-algae model

The spatiotemporal patterns of a reaction diffusion mussel-algae system with a delay subject to Neumann boundary conditions is considered. The paper is a continuation of our previous studies on delay-diffusion mussel-algae model. The global existence and positivity of solutions are obtained. The stability of the positive constant steady state and existence of Hopf bifurcation and Turing bifurcation are discussed by analyzing the distribution of eigenvalues. Furthermore, the dynamic classifications near the Turing-Hopf bifurcation point are obtained in the dimensionless parameter space by calculating the normal form on the center manifold, and the spatiotemporal patterns consisting of spatially homogeneous periodic solutions, spatially inhomogeneous steady states, and spatially inhomogeneous periodic solutions are identified in this parameter space through some numerical simulations. Both theoretical and numerical results reveal that the Turing-Hopf bifurcation can enrich the diversity of spatial distribution of populations.