Abstract
We compare spot patterns generated by Turing mechanisms with those generated by replication cascades, in a model one-dimensional reaction-diffusion system. We determine the stability region of spot solutions in parameter space as a function of a natural control parameter (feed-rate) where degenerate patterns with different numbers of spots coexist for a fixed feed-rate. While it is possible to generate identical patterns via both mechanisms, we show that replication cascades lead to a wider choice of pattern profiles that can be selected through a tuning of the feed-rate, exploiting hysteresis and directionality effects of the different pattern pathways.
Highlights
Reaction-diffusion systems are well known to self-organize into a variety of spatio-temporal patterns including, spots, stripes, spirals, as well as spatio-temporal chaos and uniform oscillations [1,2,3]
Depending on the value of the feed-rate, the system may asymptote into one of many states and the feedrate can be thought of playing the role of a natural control parameter
The relevant control parameter in this system is the feeding voltage. Another example that have attracted interest recently is found in the realm of fluid dynamics where ‘‘spots’’ of turbulent regions in pipe flow [11] and plane Couette flow [12] have been observed: On a laminar background, patches of localized turbulence, called puffs, emerge via finiteamplitude perturbations and show splitting behavior
Summary
Reaction-diffusion systems are well known to self-organize into a variety of spatio-temporal patterns including, spots, stripes, spirals, as well as spatio-temporal chaos and uniform oscillations [1,2,3]. The relevant control parameter in this system is the feeding voltage Another example that have attracted interest recently is found in the realm of fluid dynamics where ‘‘spots’’ of turbulent regions in pipe flow [11] and plane Couette flow [12] have been observed: On a laminar background, patches of localized turbulence, called puffs, emerge via finiteamplitude perturbations and show splitting behavior. These systems have been recently mapped onto excitable reactiondiffusion systems [13], and subsequently, the Turing mechanism has been proposed to explain the periodic arrangement of puffs in [14], suggesting again a reaction-diffusion framework for the dynamics. We find degeneracies, hysteresis and directionality effects that can be exploited for the purposes of pattern selection, via the tuning of the feed-rate
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