Global attractivity of equilibrium in Gierer–Meinhardt system with activator production saturation and gene expression time delays

Chen, Shanshan; Shi, Junping

doi:10.1016/j.nonrwa.2012.12.004

Cited by 24 publications

(16 citation statements)

References 55 publications

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“…The inequalities (2.5) show that (c 1 , c 2 ) and (c 1 , c 2 ) are a pair of coupled upper and lower solutions of system (1.1) as in the definition in [14,15] (see also [16]), as the nonlinearities in (1.1) are mixed quasimonotone. It is clear that there exists…”

Section: The Main Resultsmentioning

confidence: 99%

Global stability in a diffusive Holling–Tanner predator–prey model

Chen

Shi

2012

Applied Mathematics Letters

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Section: The Main Resultsmentioning

confidence: 99%

Global stability in a diffusive Holling–Tanner predator–prey model

Chen

Shi

2012

Applied Mathematics Letters

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“…Although there are now general results for the linear stability of spatially uniform steady-states under the effect of a time-delay (cf. [2]), which have been applied to the Gierer-Meinhardt (GM) model with saturated and time-delayed reaction kinetics in [1], the effect of time-delays in the reaction kinetics on localized RD patterns is not nearly as well understood.…”

Section: (Communicated By Shin-ichiro Ei)mentioning

confidence: 99%

A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model

Fadai¹,

Ward²,

Wei³

2018

Discrete &Amp; Continuous Dynamical Systems - B

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We study the spectrum of a new class of nonlocal eigenvalue problems (NLEPs) that characterize the linear stability properties of localized spike solutions to the singularly perturbed two-component Gierer-Meinhardt (GM) reaction-diffusion (RD) system with a fixed time-delay T in only the nonlinear autocatalytic activator kinetics. Our analysis of this model is motivated by the computational study of Seirin Lee et al. [Bull. Math. Bio., 72(8), (2010)] on the effect of gene expression time delays on spatial patterning for both the GM and some related RD models. For various limiting forms of the GM model, we show from a numerical study of the associated NLEP, together with an analytical scaling law analysis valid for large delay T , that a time-delay in only the activator kinetics is stabilizing in the sense that there is a wider region of parameter space where the spike solution is linearly stable than when there is no time delay. This enhanced stability behavior with a delayed activator kinetics is in marked contrast to the de-stabilizing effect on spike solutions of having a time-delay in both the activator and inhibitor kinetics. Numerical results computed from the RD system with delayed activator kinetics are used to validate the theory for the 1-D case. Introduction. For activator-inhibitor two-component reaction-diffusion (RD)systems, it is a well-known result, originating from Turing [21], that a small perturbation of a spatially uniform steady-state solution can become unstable when the diffusivity ratio is large enough. This initial instability then leads to the generation of large-amplitude stable spatial patterns. Although this mechanism for the development of spatially inhomogeneous patterns is well-understood, and has been applied to a broad range of specific RD systems (cf. [7]) and modeling scenarios on various spatial scales, what is less well-understood is the effect on pattern development of any time-delays in the reaction kinetics. Although there are now general results for the linear stability of spatially uniform steady-states under the effect of a

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“…The inequalities (6) show that (c 1 , c 2 ) and (c 1 , c 2 ) are a pair of coupled lower and upper solutions of system (1) as in the definition in [24,25](see also [26]), as the nonlinearities in (1) are mixed quasimonotone. It is clear that there exists K > 0 such that for any (c 1 , c 2 )…”

Section: The Main Resultsmentioning

confidence: 99%

“…In the another paper [8], the authors studied the stability of diffusive predator-prey model of Holling-Tanner type (3) by the construction of a standard linearization procedure and a Lyapunov function. Chen and Shi [26] focused attention on the steady states of (3). They applied the defined iteration and comparison principle sequences to prove the global asymptotic stability.…”

Section: Introductionmentioning

confidence: 99%