Unilateral regulation breaks regularity of Turing patterns

Abstract

We consider a reaction-diffusion system undergoing Turing instability and augment it by an additional unilateral source term. We investigate its influence on the Turing instability and on the character of resulting patterns. The nonsmooth positively homogeneous unilateral term τv^{-} has favorable properties, but the standard linear stability analysis cannot be performed. We illustrate the importance of the nonsmoothness by a numerical case study, which shows that the Turing instability can considerably change if we replace this term by its arbitrarily precise smooth approximation. However, the nonsmooth unilateral term and all its approximations yield qualitatively similar patterns although not necessarily developing from small disturbances of the spatially homogeneous steady state. Further, we show that the unilateral source breaks the approximate symmetry and regularity of the classical patterns and yields asymmetric and irregular patterns. Moreover, a given system with a unilateral source produces spatial patterns even for diffusion parameters with ratios closer to 1 than the same system without any unilateral term.

Unilateral sources and sinks of an activator in reaction–diffusion systems exhibiting diffusion-driven instability

“…[7] and references therein for the case of sources described by variational inequalities, [4] for unilateral sources described by multivalued maps and [3], [6] for the case of unilateral terms similar to the current paper. These results motivated numerical experiments [12] showing that for a concrete model also spatial patterns arise from small initial perturbations for diffusion parameters from D S , where it is not the case without unilateral sources. The sense of these results is positive because one of the problems of Turing's theory is that the set of diffusion parameters for which diffusion-driven instability occurs is too small, so unilateral sources for v improve this situation.…”

An influence of unilateral sources and sinks in reaction–diffusion systems exhibiting Turing’s instability on bifurcation and pattern formation

“…Also every bifurcation point (14), (8) or (13), (8) is a critical point of (19), (8) or (18), (8), respectively. To prove this implication, one needs properties of β and F we proved in Lemma 4.1 and denition of nonlinear operators corresponding to higher order terms n 1 , n 2 for which the growth conditions (17) are necessary. The proof is the same as the proof in the case of problems with unilateral terms and it can be found in appendix of [15] (see Lemma A.2).…”

“…In this section we will present a collection of results of our numerical experiments with specic reaction kinetics. In some sense we take inspiration in the paper [17], where Vejchodský et al investigated the inuence of unilateral sources of the type τ v − (and its modications) in the inhibitor equation of the reaction-diusion system on pattern formation. Analytical results for systems with unilateral terms suggest that the domain of instability could be bigger than in the classical case.…”

“…The case with unilateral terms in the activator equation again brings opposite results. There were performed many numerical experiments in [17] and [18] for the case of the unilateral source τ v − (and its modications) in the inhibitor equation for the specic reaction kinetics.…”

“…Therefore, we will use dierent, maybe more simple, approach to get our results, similar to the one in [14]. The numerical experiments are inspired by analytical results from [15] and this paper in some sense follows the paper [17]. The text is divided in the following manner.…”