Mechanism Generating Spatial Patterns in Reaction-Diffusion Systems

Suzuki, Kanako

doi:10.4036/iis.2011.131

Cited by 2 publications

(2 citation statements)

References 10 publications

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“…DDI has inspired a vast number of mathematical models since the seminal paper of Turing (1952), providing explanations of symmetry breaking and de novo pattern formation, shapes of animal coat markings, and oscillating chemical reactions. We refer the reader to the monographs by Murray (2002, 2003) and to the review article (Suzuki 2011) for references on DDI in the two component reaction-diffusion systems and to the paper Satnoianu et al. (2000) in the several component systems.…”

Section: Introductionmentioning

confidence: 99%

Instability of turing patterns in reaction-diffusion-ODE systems

2016

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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-

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Section: Introductionmentioning

confidence: 99%

Instability of turing patterns in reaction-diffusion-ODE systems

2016

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“…ε > 0 is small) while the inhibitor diffuses rapidly (D > 0 is large) to generate patterns. [6,15,30,31,33,[41][42][43] In the shadow type reduction of this system…”

Section: Remark 11 (Numerical Simulations Of the Model)mentioning

confidence: 99%

Dynamical spike solutions in a nonlocal model of pattern formation

et al. 2018

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Coupling a reaction-diffusion equation with ordinary differential equations (ODE) may lead to diffusion-driven instability (DDI) which, in contrast to the classical reaction-diffusion models, causes destabilization of both, constant solutions and Turing patterns. Using a shadow-type limit of a reaction-diffusion-ODE model, we show that in such cases the instability driven by nonlocal terms (a counterpart of DDI) may lead to formation of unbounded spike patterns.

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Mechanism Generating Spatial Patterns in Reaction-Diffusion Systems

Cited by 2 publications

References 10 publications

Instability of turing patterns in reaction-diffusion-ODE systems

Instability of turing patterns in reaction-diffusion-ODE systems

Dynamical spike solutions in a nonlocal model of pattern formation

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