Delayed Reaction Kinetics and the Stability of Spikes in the Gierer--Meinhardt Model

Nabil T Fadai,Michael J Ward,Juncheng Wei

doi:10.1137/16m1063460

Abstract

A linear stability analysis of localized spike solutions to the singularly perturbed two-component Gierer--Meinhardt (GM) reaction-diffusion (RD) system with a fixed time delay $T$ in the nonlinear reaction kinetics is performed. Our analysis of this model is motivated by the computational study of Lee, Gaffney, and Monk [Bull. Math. Bio., 72 (2010), pp. 2139--2160] on the effect of gene expression time delays on spatial patterning for both the GM model and some related RD models. It is shown that the linear stability properties of such localized spike solutions are characterized by the discrete spectra of certain nonlocal eigenvalue problems (NLEP). Phase diagrams consisting of regions in parameter space where the steady-state spike solution is linearly stable are determined for various limiting forms of the GM model in both 1-dimensional and 2-dimensional domains. On the boundary of the region of stability, the spike solution is found to undergo a Hopf bifurcation. For a special range of exponents in the nonlinearities for the 1-dimensional GM model, and assuming that the time delay only occurs in the inhibitor kinetics, this Hopf bifurcation boundary is readily determined analytically. For this special range of exponents, the challenging problem of locating the discrete spectrum of the NLEP is reduced to the much simpler problem of locating the roots to a simple transcendental equation in the eigenvalue parameter. By using a hybrid analytical-numerical method, based on a parametrization of the NLEP, it is shown that qualitatively similar phase diagrams occur for general GM exponent sets and for the more biologically relevant case where the time delay occurs in both the activator and inhibitor kinetics. Overall, our results show that there is a critical value $T_{\star}$ of the delay for which the spike solution is unconditionally unstable for $T>T_{*}$, and that the parameter region where linear stability is assured is, in general, rather limited. A comparison of the theory with full numerical results computed from the RD system with delayed reaction kinetics for a particular parameter set suggests that the Hopf bifurcation can be subcritical, leading to a global breakdown of a robust spatial patterning mechanism.

Highlights

In [21], Turing proposed that localized peaks in the concentration of a chemical substance, known as a morphogen, could be responsible for the process of morphogenesis, which describes the development of a complex organism from a single cell
The main goal of this paper is to provide a theoretical framework to predict parameter ranges where stable spatial patterning exists for this GM model of [14] with delayed reaction kinetics
Motivated by the computational studies of pattern formation in RD systems with a time delay in the reaction kinetics, by modeling gene expression time delays, we have analyzed the linear stability of spike solutions to various limiting forms of the GM RD model with delayed reaction kinetics in both 1-D and 2-D domains

Summary

Introduction

In [21], Turing proposed that localized peaks in the concentration of a chemical substance, known as a morphogen, could be responsible for the process of morphogenesis, which describes the development of a complex organism from a single cell. Motivated by the previous computational studies (cf [5], [14], [15], [16]) of spatial patterning for the GM and related models with delayed reaction kinetics, the main goal of this paper is to analyze the linear stability of steady-state spike solutions for (1.1) and its two-dimensional (2-D) counterpart to O(1) time scale instabilities when the reaction kinetics have a time delay T. Any such instability is an instability of the amplitude of the spike, and is associated with unstable O(1) eigenvalues in an NLEP.

Delay effects

Inhibitor delay

Inhibitor delay effects

Discussion