Reaction and Diffusion on Growing Domains: Scenarios for Robust Pattern Formation

Abstract

We investigate the sequence of patterns generated by a reaction-diffusion system on a growing domain. We derive a general evolution equation to incorporate domain growth in reaction-diffusion models and consider the case of slow and isotropic domain growth in one spatial dimension. We use a self-similarity argument to predict a frequency-doubling sequence of patterns for exponential domain growth and we find numerically that frequency-doubling is realized for a finite range of exponential growth rate. We consider pattern formation under different forms for the growth and show that in one dimension domain growth may be a mechanism for increased robustness of pattern formation.

Pattern Formation in Reaction–Diffusion Models with Nonuniform Domain Growth

“…3). This is similar to the asymmetries observed for high strain rates for uniform domain growth reported in Crampin et al (1999). This represents a smooth modulation of the frequency-doubling sequence.…”

“…We have shown that for small spatial perturbation, which we interpret as gentle gradients in the spatial dependence, we recover mode-doubling-type sequences (at least for the low modes that we have monitored). For more strongly nonuniform growth (for steeper change in S) at high ρ we have observed asymmetry, as for the uniform case, and for lower ρ selected peaks fail to undergo transitions, akin to the failure of mode-doubling for uniform linear growth (Crampin et al, 1999). For highly nonuniform domain growth, in particular when one region of the domain is not growing, other sequences may be formed by the failure of peak splitting (or insertion) at particular locations, specifically for those peaks on the stationary part.…”

“…4). This defect in the peak splitting is reminiscent of the breakdown of the frequencydoubling sequence at very low uniform strain rates (Crampin et al, 1999), however, † We note that in the figures the patterns are shown in the transformed variables on the unit interval. This is for ease of comparison between different pattern sequences.…”

“…Below we present solutions for constant strain rate S = ρ [see Crampin et al (1999) for results with time-dependent strain rate]. In this case the evolution equations are classical reaction-diffusion modified by a constant dilution term and time-dependent diffusivity.…”

“…Under certain circumstances these mechanisms can be made to operate simultaneously [mode-tripling, see Crampin et al (2002)]. Patterns evolving on the growing domain follow a sequence which depends on the first pattern to form but is otherwise insensitive to initial conditions, and hence it has been suggested that this is a mechanism for reliable pattern formation (Crampin et al, 1999).…”