The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system

Yi, Fengqi; Gaffney, Eamonn A.; Seirin-Lee, Sungrim

doi:10.3934/dcdsb.2017031

Cited by 23 publications

(29 citation statements)

References 38 publications

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“…It is observed that the delay can considerably increase the time required for the formation of a stable pattern, while on rapidly growing domains the delay may result in the absence of a pattern. Further studies, again using models with discrete time-delays, reinforce the observation that structural robustness has a complicated dependence on the delay [36][37][38][39]. It would be of considerable interest to investigate TP models with distributed delays, which are the more realistic continuous analogue of discrete delays, to ascertain their effect on structural robustness.…”

Section: Design Principles Of Robust Turing Networkmentioning

confidence: 73%

Turing pattern design principles and their robustness

Vittadello

Leyshon

Schnoerr

et al. 2021

Phil. Trans. R. Soc. A.

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Turing patterns have morphed from mathematical curiosities into highly desirable targets for synthetic biology. For a long time, their biological significance was sometimes disputed but there is now ample evidence for their involvement in processes ranging from skin pigmentation to digit and limb formation. While their role in developmental biology is now firmly established, their synthetic design has so far proved challenging. Here, we review recent large-scale mathematical analyses that have attempted to narrow down potential design principles. We consider different aspects of robustness of these models and outline why this perspective will be helpful in the search for synthetic Turing-patterning systems. We conclude by considering robustness in the context of developmental modelling more generally. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

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Section: Design Principles Of Robust Turing Networkmentioning

confidence: 73%

Turing pattern design principles and their robustness

Vittadello

Leyshon

Schnoerr

et al. 2021

Phil. Trans. R. Soc. A.

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“…(1) On the basis of results of [34], we have obtained a much larger range where Turing instability does not occur, which is one sufficient and necessary condition. In other words, we give the weaker conditions that guarantee Turing instability.…”

Section: Introductionmentioning

confidence: 86%

“…(1−ε)π 2 , 0 < ε < 1} ⊆ {(d, ε) : ε > ε B (d), d > 0}, which means that we have given a much larger range, where Turing instability does not occur, than[34]. (See Theorem 3.1 (1) of[34] ).Remark 2.7. We call the critical curve of Turing instability ε = ε * (d), d > 0 the first Turing bifurcation curve, on which the corresponding characteristic equation without delay has no root with positive real part.…”

mentioning

confidence: 98%

Turing Instability and Turing–Hopf Bifurcation in Diffusive Schnakenberg Systems with Gene Expression Time Delay

Jiang¹,

Wang²,

Cao³

2018

J Dyn Diff Equat

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For delayed reaction-diffusion Schnakenberg systems with Neumann boundary conditions, critical conditions for Turing instability are derived, which are necessary and sufficient. And existence conditions for Turing, Hopf and Turing-Hopf bifurcations are established. Normal forms truncated to order 3 at Turing-Hopf singularity of codimension 2, are derived. By investigating Turing-Hopf bifurcation, the parameter regions for the stability of a periodic solution, a pair of spatially inhomogeneous steady states and a pair of spatially inhomogeneous periodic solutions, are derived in (τ, ε) parameter plane (τ for time delay, ε for diffusion rate). It is revealed that joint effects of diffusion and delay can lead to the occurrence of mixed spatial and temporal patterns. Moreover, it is also demonstrated that various spatially inhomogeneous patterns with different spatial frequencies can be achieved via changing the diffusion rate. And, the phenomenon that time delay may induce a failure of Turing instability observed by Gaffney and Monk (2006) are theoretically explained.

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“…Another significant aspect causing lots of focuses is the property of solutions for this model with different boundary conditions [9], [2]. Furthermore, regarding the pattern formation of reaction-diffusion equations, it is of a high value to consider the major effects of the strength or means of diffusion and reaction on their patterns [1], [22], [3], [21], [6].…”

Section: Introductionmentioning

confidence: 99%

Dynamics and patterns of an activator-inhibitor model with cubic polynomial source

Li¹,

Jiang²

2019

Appl.Math.

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The dynamics of an activator-inhibitor model with general cubic polynomial source is investigated. Without diffusion, we consider the existence, stability and bifurcations of equilibria by both eigenvalue analysis and numerical methods. For the reactiondiffusion system, a Lyapunov functional is proposed to declare the global stability of constant steady states, moreover, the condition related to the activator source leading to Turing instability is obtained in the paper. In addition, taking the production rate of the activator as the bifurcation parameter, we show the decisive effect of each part in the source term on the patterns and the evolutionary process among stripes, spots and mazes. Finally, it is illustrated that weakly linear coupling in the activator-inhibitor model can cause synchronous and anti-phase patterns.

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The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system

Cited by 23 publications

References 38 publications

Turing pattern design principles and their robustness

Turing pattern design principles and their robustness

Turing Instability and Turing–Hopf Bifurcation in Diffusive Schnakenberg Systems with Gene Expression Time Delay

Dynamics and patterns of an activator-inhibitor model with cubic polynomial source

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