Stochastic reaction and diffusion on growing domains: understanding the breakdown of robust pattern formation
All biological patterns, from population densities to animal coat markings, can be thought of as heterogeneous spatio-temporal distributions of mobile agents. Many mathematical models have been proposed to account for the emergence of this complexity, but, in general, they have consisted of deterministic systems of differential equations, which do not take into account the stochastic nature of population interactions. One particular, pertinent, criticism of these deterministic systems, is that the exhibited patterns can often be highly sensitive to changes in initial conditions, domain geometry, parameter values, etc. Due to this sensitivity, we seek to understand the effects of stochasticity and growth on paradigm biological patterning models. We extend spatial Fourier analysis and growing domain mapping techniques to encompass stochastic Turing systems. Through this we find that the stochastic systems are able to realise much richer dynamics than their deterministic counter parts, in that patterns are able to exist outside the standard Turing parameter range. Further, it is seen that the inherent stochasticity in the reactions appears to be more important than the noise generated by growth, when considering which wave modes are excited. Finally, although growth is able to generate robust pattern sequences in the deterministic case, we see that stochastic effects destroy this mechanism of pattern doubling. However, through Fourier analysis we are able to suggest a reason behind this lack of robustness and identify possible mechanisms by which to reclaim it.