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

Abstract

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

Modern perspectives on near-equilibrium analysis of Turing systems

“…However, these morphogens are often, though not always, signalling molecules which must be produced via gene transcription, leading to delays between uptake of a signal and the up-or downregulation of the resultant product. Such time delays have been studied in the past few decades [129,[198][199][200][201][202], broadly with the conclusion that they may lead to oscillatory behaviours, rather than the formation of stationary patterns (though see [203] for a case where delay can enhance the stability of spikes). Such results plausibly call into question the standard use of reaction-diffusion frameworks for the fastest cellular self-organization processes, where oscillations are not typically observed, though we caution that there is more work to be done both biologically and mathematically in testing these hypotheses.…”

Modern Perspectives on Near-Equilibrium Analysis of Turing Systems

“…However, these morphogens are often, though not always, signalling molecules which must be produced via gene transcription, leading to delays between uptake of a signal and the up or down regulation of the resultant product. Such time delays have been studied in the past few decades [54,62,63,92,121,211], broadly with the conclusion that they may lead to oscillatory behaviours, rather than the formation of stationary patterns (though see [55] for a case where delay can enhance the stability of spikes). Such results plausibly call into question the standard use of reaction-diffusion frameworks for the fastest cellular self-organisation processes, where oscillations are not typically observed, though we caution that there is more work to be done both biologically and mathematically in testing these hypotheses.…”

Stability and dynamics of spike-type solutions to delayed Gierer-Meinhardt equations

“…We repeated the process for different values of L and similar results are found in each case. In [5] it is found that this form of delay can actually aid in the stabilization of spike solutions. Simulations of (36) with delay values of up to 50 resulted in stable solutions with no oscillations.…”