Stochastic Turing Patterns

The problem of explaining the rich spectrum of different patterns exhibited by mammals has a long history. In a controversial paper [Turing52], Turing proposed a Reaction-Diffusion (RD) model to explain at a macroscopic scale the process of pattern formation, as related to the occurrence of what he called a diffusion-driven instability. A typical Turing system consists of at least two chemical species, usually referred to as activator and inhibitor, reacting in such a way that their steady state is stable to small perturbations in the absence of diffusion, but becomes unstable when diffusion is present. This simple mechanism (also referred to as Turing instability) was indicated by Turing as being responsible for the creation of spatial patterns in mammals [Murray93]. The traditional model is deterministic (although it is diffusion based!) In the sense that a stable pattern will not evolve. The assumption of infinitely many molecules that can be represented by frequencies leads to partial differential equations. This project proposes to explore stochastic partial differential equation analogues.