Unilateral regulation breaks regularity of Turing patterns

TL;DR

This paper studies how adding a unilateral source term to reaction-diffusion systems affects Turing patterns, leading to irregular, asymmetric patterns and changing the conditions for pattern formation.

Contribution

It introduces a nonsmooth unilateral term into reaction-diffusion models and analyzes its impact on pattern regularity and stability, highlighting differences from smooth approximations.

Findings

Unilateral source induces irregular, asymmetric patterns.

Patterns can form with diffusion ratios closer to 1.

Nonsmoothness significantly influences pattern characteristics.

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 $\tau v^-$ has favourable 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.