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

Abstract

We study two-species reaction–diffusion systems on growing manifolds, including situations where the growth is anisotropic yet dilational in nature. In contrast to the literature on linear instabilities in such systems, we study how growth and anisotropy impact the qualitative properties of nonlinear patterned states which have formed before growth is initiated. We produce numerical solutions to numerous reaction–diffusion systems with varying reaction kinetics, manner of growth (both isotropic and anisotropic), and timescales of growth on both planar elliptical and curved ellipsoidal domains. We find that in some parameter regimes, some of these factors have a negligible effect on the long-time patterned state. On the other hand, we find that some of these factors play a role in determining the patterns formed on surfaces and that anisotropic growth can produce qualitatively different patterns to those formed under isotropic growth.

Diffusive instabilities and spatial patterning from the coupling of reaction–diffusion processes with Stokes flow in complex domains

“…1999; Plaza et al. 2004; Krause, Ellis & Van Gorder 2019). Other related topics which may merit consideration include patterning on growing domains immersed in fluid baths, or patterning of chemicals in thin films on the surface of growing domains.…”

“…In such cases, for appropriate boundary data and basin geometries, one can obtain flows of the form A = (A(y, t), 0). Interestingly, related mathematical problems have already been considered in the setting of the Turing instability and pattern formation on growing domains: from conservation of mass and applying Reynold's transport theorem within a growing domain, a time-dependent advection term appears in the reaction diffusion equations, much like the advective velocity field arising in a time-dependent flow (Crampin et al 1999;Plaza et al 2004;Krause, Ellis & Van Gorder 2019). Other related topics which may merit consideration include patterning on growing domains immersed in fluid baths, or patterning of chemicals in thin films on the surface of growing domains.…”

Turing Patterning in Stratified Domains

“…In such cases, approximate solutions for the system's eigenfunctions need to be derived that are orthogonal in the patterning layer. Examples that deviate even further from the classical case are growing domains (Crampin et al 1999;Plaza et al 2004;Krause et al 2019;Sánchez-Garduño et al 2019) and spatially heterogeneous reaction-diffusion processes (Benson et al 1998;Page et al 2003Page et al , 2005Haim et al 2015;Kolokolnikov and Wei 2018), for which the canonical approach does not work. In such cases, novel approaches to pattern-forming instabilities have recently been developed for growth (Madzvamuse et al 2010;) and heterogeneity (Krause et al 2020) under certain simplifications, but such analyses are quite different to the classical case.…”

“… 2004 ; Krause et al. 2019 ; Sánchez-Garduño et al. 2019 ) and spatially heterogeneous reaction–diffusion processes (Benson et al.…”

Global Existence of Solutions to Reaction-Diffusion Systems with Mass Transport Type Boundary Conditions

“…Again these models arise in the area of tissue engineering and regenerative medicine, elctrospun membrane which are useful in applications such as filtration systems and sensors for chemical detection. In [4], [24], [13] and the references therein, the authors derive the equation for the reaction diffusion equation on a growing manifold with or without boundary. They imposed special growth conditions such as isotropic (including exponential) or anisotropic and studied the behaviour of solutions.…”