Stochastic Patterns and the Role of Crowding

A stochastic variant of the Brusselator model is investigated. The model accounts for a long range coupling among constituents, as well as for the finite capacity of the embedding medium. The mean field limit of the model is studied and the condition for Turing and wave instability obtained. A degenerate, cusp like transition that separates the domains of Turing and wave order can take place. The point of transition is worked out analytically. Interestingly, the region of Turing instability, as delimited by such transition point, can set in also if the inhibitor diffuses slower then the activator. This is a consequence of the generalized diffusion scheme here analyzed and which originates from having imposed an effect of spatial competition. Beyond the deterministic, mean field picture, we elaborate on the role of stochastic corrections. Granularity, endogenous to the system, can eventually materialize in waves or Turing like patterns, that we here categorized in distinct classes.