Turing–Hopf patterns on growing domains: The torus and the sphere

Sánchez‐Garduño, Faustino; Krause, Andrew L.; Castillo, Jorge A.; Padilla, Pablo

doi:10.1016/j.jtbi.2018.09.028

Cited by 40 publications

(38 citation statements)

References 60 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As far as we are aware, no current literature has considered diffusive instabilities inside of a torus, though some authors have considered pattern forming systems on the surface of a torus (Sánchez-Garduño et al. 2018). Fluid flows in tori are especially of interest in the mathematical side of hydrodynamics, as such a geometry is non-trivial but still suitable for some analytical insight (Chen & Price 1996).…”

Section: Pattern Formation In More Complex Geometriesmentioning

confidence: 99%

“…a simple yet non-trivial example of circulating flow in a curved geometry which has been exploited in combustion research (Nenniger et al 1985), and recent bioreactor experiments have been designed which result in toroidal geometries (Vadivelu et al 2017). As far as we are aware, no current literature has considered diffusive instabilities inside of a torus, though some authors have considered pattern forming systems on the surface of a torus (Sánchez-Garduño et al 2018). Fluid flows in tori are especially of interest in the mathematical side of hydrodynamics, as such a geometry is non-trivial but still suitable for some analytical insight (Chen & Price 1996).…”

Section: Three-dimensional Toroidal Domainmentioning

confidence: 99%

See 1 more Smart Citation

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

2019

Self Cite

View full text Add to dashboard Cite

We study spatial and spatio-temporal pattern formation emergent from reaction–diffusion–advection systems formed by considering reaction–diffusion systems coupled to prescribed fluid flows. While there have been a number of studies on the planar dynamics of such systems and the resulting instabilities and spatio-temporal patterning in the plane, less has been done on complicated flows in complex domains. We consider a general approach for the study of bounded domains in order to model two- and three-dimensional geometries which are more likely to be of relevance for modelling dynamics within fluid vessels used in experiments. Considering a variety of problem geometries with finite cross-sections, such as two-dimensional channels, three-dimensional ducts and three-dimensional pipes, we demonstrate the role cross-section geometry plays in pattern formation under such systems. We find that the generic instability is that of an oscillatory or wave Turing instability, resulting in patterns which change in time, often being advected with the fluid flow. As in previous works, we observe a change in patterns formed when progressing from zero to weak to strong advection for uniform advection across the domain, with particularly strong advection destroying patterns. One novel finding is that heterogeneous fluid flow can induce qualitatively different patterns across the domain. For instance, Poiseuille flow with maximal advection in the centre of a vessel and zero advection at the boundary of a vessel is shown to exhibit patterns in the centre of the vessel which are different from patterns near the boundary, with differences attributed to the differential local advection within each region of the vessel. Additionally, we observe sheared patterns, which appear due to gradients in the fluid velocity, and cannot be obtained via any kind of uniform flow. Finally we also explore flow in more complex domains, including wavy-walled channels, continuous stirred-tank reactors, U-shaped pipes and a toroidal domain, in order to demonstrate behaviours when the flow is both heterogeneous and bidirectional, as well as to demonstrate that our results still apply for complex finite domains. Our analysis suggests that such non-trivial advection results in moving patterns which are more complex than observed in simpler reaction–diffusion–advection, and may be more characteristic of realistic flow regimes in biological media.

show abstract

Section: Pattern Formation In More Complex Geometriesmentioning

confidence: 99%

Section: Three-dimensional Toroidal Domainmentioning

confidence: 99%

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

2019

Self Cite

View full text Add to dashboard Cite

show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%

“… 2019 ; Sánchez-Garduño et al. 2019 ) and spatially heterogeneous reaction–diffusion processes (Benson et al. 1998 ; Page et al.…”

Section: Introductionmentioning

confidence: 99%

Turing Patterning in Stratified Domains

et al. 2020

Self Cite

View full text Add to dashboard Cite

Reaction–diffusion processes across layered media arise in several scientific domains such as pattern-forming E. coli on agar substrates, epidermal–mesenchymal coupling in development, and symmetry-breaking in cell polarization. We develop a modeling framework for bilayer reaction–diffusion systems and relate it to a range of existing models. We derive conditions for diffusion-driven instability of a spatially homogeneous equilibrium analogous to the classical conditions for a Turing instability in the simplest nontrivial setting where one domain has a standard reaction–diffusion system, and the other permits only diffusion. Due to the transverse coupling between these two regions, standard techniques for computing eigenfunctions of the Laplacian cannot be applied, and so we propose an alternative method to compute the dispersion relation directly. We compare instability conditions with full numerical simulations to demonstrate impacts of the geometry and coupling parameters on patterning, and explore various experimentally relevant asymptotic regimes. In the regime where the first domain is suitably thin, we recover a simple modulation of the standard Turing conditions, and find that often the broad impact of the diffusion-only domain is to reduce the ability of the system to form patterns. We also demonstrate complex impacts of this coupling on pattern formation. For instance, we exhibit non-monotonicity of pattern-forming instabilities with respect to geometric and coupling parameters, and highlight an instability from a nontrivial interaction between kinetics in one domain and diffusion in the other. These results are valuable for informing design choices in applications such as synthetic engineering of Turing patterns, but also for understanding the role of stratified media in modulating pattern-forming processes in developmental biology and beyond.

show abstract

“…The system is obtained as a simplification for the Hodgkin-Huxley model describing nerve impulse propagation. Due to the essential feature to describe the initiation and propagation of action potentials in neurons [12], the FN system has been investigated from different angles including bifurcation [12,8,10,17,27], traveling wave solutions [1,3,4,7,25,26] and other dynamic aspects [2,23]. Schonbek [23] studies the local and global existence of solutions for the FN equations.…”

mentioning

confidence: 99%

Stabilities and dynamic transitions of the Fitzhugh-Nagumo system

Xing¹,

Pan²,

Wang³

2021

Discrete &Amp; Continuous Dynamical Systems - B

View full text Add to dashboard Cite

The article aims to examine the dynamic transition of the reactiondiffusion Fitzhugh-Nagumo system defined on a thin spherical shell and a 2Drectangular domain. The mathematical tool employed is the theory of phase transition dynamics established for dissipative dynamical systems. The main results in this paper include two parts. First, for the system on a thin spherical shell, we only focus on the transition from a real simple eigenvalue. More precisely, if the first eigenspace is three-dimensional, the system undergoes either a continuous transition or a jump transition. Besides, a mix transition is also allowed if the first eigenspace is one-dimensional. Second, for the system on a rectangular domain, both the transitions from a simple real eigenvalue and a pair of simple complex eigenvalues are considered. Our results imply that two steady-state solutions bifurcate, which are either attractors or saddle points, and a Hopf bifurcation is also possible in the system on the rectangular domain.

show abstract

Turing–Hopf patterns on growing domains: The torus and the sphere

Cited by 40 publications

References 60 publications

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

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

Turing Patterning in Stratified Domains

Stabilities and dynamic transitions of the Fitzhugh-Nagumo system

Contact Info

Product

Resources

About