Nonlinear stability analyses of Turing patterns for a mussel-algae model

A particular interaction-diffusion mussel-algae model system for the development of spontaneous stationary young mussel bed patterning on a homogeneous substrate covered by a quiescent marine layer containing algae as a food source is investigated employing weakly nonlinear diffusive instability analyses. The main results of these analyses can be represented by plots in the ratio of mussel motility to algae lateral diffusion versus the algae reservoir concentration dimensionless parameter space. Regions corresponding to bare sediment and mussel patterns consisting of rhombic or hexagonal arrays and isolated clusters of clumps or gaps, an intermediate labyrinthine state, and homogeneous distributions of low to high density may be identified in this parameter space. Then those Turing diffusive instability predictions are compared with both relevant field and laboratory experimental evidence and existing numerical simulations involving differential flow migrating band instabilities for the associated interaction-dispersion-advection mussel-algae model system as well as placed in the context of the results from some recent nonlinear pattern formation studies.