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The spatiotemporal patterns of a reaction diffusion mussel-algae system witha delay subject to Neumann boundary conditions is considered. The paper is acontinuation of our previous studies on delay-diffusion mussel-algae model. Theglobal existence and positivity of solutions are obtained. The stability of thepositive constant steady state and existence of Hopf bifurcation and Turingbifurcation are discussed by analyzing the distribution of eigenvalues.Furthermore, the dynamic classifications near the Turing-Hopf bifurcation pointare obtained in the dimensionless parameter space by calculating the normalform on the center manifold, and the spatiotemporal patterns consisting ofspatially homogeneous periodic solutions, spatially inhomogeneous steadystates, and spatially inhomogeneous periodic solutions are identified in thisparameter space through some numerical simulations. Both theoretical andnumerical results reveal that the Turing-Hopf bifurcation can enrich thediversity of spatial distribution of populations.

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