Spike patterns in a reaction–diffusion ODE model with Turing instability

Härting, Steffen; Marciniak–Czochra, Anna

doi:10.1002/mma.2899

Cited by 20 publications

(17 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We prove that bistability without the hysteresis effect is not sufficient for existence of stable spatially heterogenous patterns. Moreover, we provide a systematic description of stationary solutions that may differ from those of the usual reaction-diffusion systems [20,19,7]. In particular, we show a co-existence of infinite number of stationary solutions that exhibit jump-discontinuities in non-diffusing variables.…”

mentioning

confidence: 87%

Hysteresis-driven pattern formation in reaction-diffusion-ODE systems

Köthe¹,

Marciniak–Czochra²,

Izumi³

2020

Discrete &Amp; Continuous Dynamical Systems - A

Self Cite

View full text Add to dashboard Cite

mentioning

confidence: 87%

Hysteresis-driven pattern formation in reaction-diffusion-ODE systems

Köthe¹,

Marciniak–Czochra²,

Izumi³

2020

Discrete &Amp; Continuous Dynamical Systems - A

Self Cite

View full text Add to dashboard Cite

“…A variety of non-Turing patterns arising in systems coupling one diffusing component with a non-diffusing subsystem has been recently shown in Refs. [ 29 – 31 ].…”

Section: Introductionmentioning

confidence: 99%

Post-Turing tissue pattern formation: Advent of mechanochemistry

et al. 2018

Self Cite

View full text Add to dashboard Cite

Chemical and mechanical pattern formation is fundamental during embryogenesis and tissue development. Yet, the underlying molecular and cellular mechanisms are still elusive in many cases. Most current theories assume that tissue development is driven by chemical processes: either as a sequence of chemical patterns each depending on the previous one, or by patterns spontaneously arising from specific chemical interactions (such as “Turing-patterns”). Within both theories, mechanical patterns are usually regarded as passive by-products of chemical pre-patters. However, several experiments question these theories, and an increasing number of studies shows that tissue mechanics can actively influence chemical patterns during development. In this study, we thus focus on the interplay between chemical and mechanical processes during tissue development. On one hand, based on recent experimental data, we develop new mechanochemical simulation models of evolving tissues, in which the full 3D representation of the tissue appears to be critical for obtaining a realistic mechanochemical behaviour. The presented modelling approach is flexible and numerically studied using state of the art finite element methods. Thus, it may serve as a basis to combine simulations with new experimental methods in tissue development. On the other hand, we apply the developed approach and demonstrate that even simple interactions between tissue mechanics and chemistry spontaneously lead to robust and complex mechanochemical patterns. Especially, we demonstrate that the main contradictions arising in the framework of purely chemical theories are naturally and automatically resolved using the mechanochemical patterning theory.

show abstract

“…(2013), it may happen that there exist no stable stationary patterns and the emerging spatially heterogeneous structures are of a dynamical nature. In numerical simulations of such models, solutions having the form of unbounded periodic or irregular spikes have been observed (Härting and Marciniak-Czochra 2014). …”

Section: Introductionmentioning

confidence: 99%

Instability of turing patterns in reaction-diffusion-ODE systems

2016

Self Cite

View full text Add to dashboard Cite

The aim of this paper is to contribute to the understanding of the pattern formation phenomenon in reaction-diffusion equations coupled with ordinary differential equations. Such systems of equations arise, for example, from modeling of interactions between cellular processes such as cell growth, differentiation or transformation and diffusing signaling factors. We focus on stability analysis of solutions of a prototype model consisting of a single reaction-diffusion equation coupled to an ordinary differential equation. We show that such systems are very different from classical reaction-diffusion models. They exhibit diffusion-driven instability (turing instability) under a condition of autocatalysis of non-diffusing component. However, the same mechanism which destabilizes constant solutions of such models, destabilizes also all continuous spatially heterogeneous stationary solutions, and consequently, there exist no stable Turing patterns in such reaction-diffusion-ODE systems. We provide a rigorous result on the nonlinear instability, which involves the analysis of a continuous spectrum of a linear operator induced by the lack of diffusion in the desta-

show abstract

Spike patterns in a reaction–diffusion ODE model with Turing instability

Cited by 20 publications

References 20 publications

Hysteresis-driven pattern formation in reaction-diffusion-ODE systems

Hysteresis-driven pattern formation in reaction-diffusion-ODE systems

Post-Turing tissue pattern formation: Advent of mechanochemistry

Instability of turing patterns in reaction-diffusion-ODE systems

Contact Info

Product

Resources

About