  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.
