History dependence and the continuum approximation breakdown: the impact of domain growth on Turing’s instability

Abstract

A diffusively driven instability has been hypothesized as a mechanism to drive spatial self-organization in biological systems since the seminal work of Turing. Such systems are often considered on a growing domain, but traditional theoretical studies have only treated the domain size as a bifurcation parameter, neglecting the system non-autonomy. More recently, the conditions for a diffusively driven instability on a growing domain have been determined under stringent conditions, including slow growth, a restriction on the temporal interval over which the prospect of an instability can be considered and a neglect of the impact that time evolution has on the stability properties of the homogeneous reference state from which heterogeneity emerges. Here, we firstly relax this latter assumption and observe that the conditions for the Turing instability are much more complex and depend on the history of the system in general. We proceed to relax all the above constraints, making analytical progress by focusing on specific examples. With faster growth, instabilities can grow transiently and decay, making the prediction of a prospective Turing instability much more difficult. In addition, arbitrarily high spatial frequencies can destabilize, in which case the continuum approximation is predicted to break down.

Bespoke Turing Systems

“…( 2020b ), Kozák et al. ( 2019 ) for spatial heterogeneity and Klika and Gaffney ( 2017 ), Madzvamuse et al. ( 2010 ), Van Gorder ( 2020 ) for temporal forcing.…”

Influence of Curvature, Growth, and Anisotropy on the Evolution of Turing Patterns on Growing Manifolds

“…Firstly, the growth rate and kind of growth (e.g., isotropic or anisotropic) can significantly change the final pattern, even though we can demonstrate that the final pattern is a steady state. This suggests that the high level of multistability in these systems on fixed domains can be heavily influenced by their history due to growth, as suggested by Klika and Gaffney ( 2017 ). However, the functional form of growth was never really important; all simulations using different forms of the growth rates were qualitatively the same with linear growth at a suitable rate, as long as the initial and final manifolds were fixed.…”

“…Such conditions have been extended to time-dependent domains (Madzvamuse et al. 2010 ; Klika and Gaffney 2017 ), as well as to isotropically growing manifolds (Plaza et al. 2004 ).…”

Periodic solutions and chaos in the Barkley pipe model on a finite domain

“…While invoking SIBC, it is observed that solutions depend strongly on the length of the numerical domain N , in addition to the parameters r and τ . As the domain size N impacts the nature of the solutions, it may be treated as an implicit bifurcation parameter (see [18] for a treatment of domain length as an explicit bifurcation parameter). Thus the variable bifurcation parameters are r(N ) and τ (N ).…”