Moving finite element simulations for reaction-diffusion systems

2011-2012 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe

The Gierer-Meinhardt system on a compact two-dimensional Riemannian Manifold: Interaction of Gaussian curvature and Green's function

In this paper, we rigorously prove the existence and stability of single-peaked patterns for the singularly perturbed Gierer-Meinhardt system on a compact two-dimensional Riemannian manifold without boundary which are far from spatial homogeneity. Throughout the paper we assume that the activator diffusivity is small enough. We show that for a threshold ratio of the activator diffusivity and the inhibitor diffusivity, the Gaussian curvature and the Green's function interact. A convex combination of the Gaussian curvature and the Green's function together with their derivatives are linked to the peak locations and the o(1) eigenvalues. A nonlocal eigenvalue problem (NLEP) determines the O(1) eigenvalues which all have negative part in this case.RGC of Hong Kon

On the Gierer-Meinhardt system with precursors

We consider the Gierer-Meinhardt system with a for the activator. Such an equation exhibits a typical Turing bifurcation of the second kind, i.e., homogeneous uniform steady states do not exist in the system. We establish the existence and stability of N-peaked steady-states in terms of the precursor and the inhibitor diffusivity. It is shown that the precursor plays an essential role for both existence and stability of spiky patterns. In particular, we show that precursors can give rise to instability. This is a new effect which is not present in the homogeneous case

Center manifold reduction approach for the lubrication equation

The goal of this study is the reduction of the lubrication equation, modelling thin film dynamics, onto an approximate invariant manifold.~The reduction is derived for the physical situation of the late phase evolution of a dewetting thin liquid film, where arrays of droplets connected by an ultrathin film of thickness \eps undergo a slow-time coarsening dynamics.~With this situation in mind, we construct an asymptotic approximation of the corresponding invariant manifold, that is parametrized by a family of droplet pressures and positions, in the limit when \eps\to 0. The approach is inspired by the paper by Mielke and Zelik [Mem. Amer. Math. Soc., Vol. 198, 2009], where the center manifold reduction was carried out for a class of semilinear systems. In this study this approach is considered for quasilinear { degenerate }parabolic PDE's such as lubrication equations. While it has previously been shown by Glasner and Witelski [Phys. Rev. E, Vol. 67, 2003], that the system of ODEs governing the coarsening dynamics, can be obtained via formal asymptotic methods, the center manifold reduction approach presented here, pursues the rigorous justification of this asymptotic limit

Slowly varying control parameters, delayed bifurcations and the stability of spikes in reaction-diffusion systems

We present three examples of delayed bifurcations for spike solutions of reaction-diffusion systems. The delay effect results as the system passes slowly from a stable to an unstable regime, and was previously analysed in the context of ODE's in [P.Mandel, T.Erneux, J.Stat.Phys, 1987]. It was found that the instability would not be fully realized until the system had entered well into the unstable regime. The bifurcation is said to have been "delayed" relative to the threshold value computed directly from a linear stability analysis. In contrast, we analyze the delay effect in systems of PDE's. In particular, for spike solutions of singularly perturbed generalized Gierer-Meinhardt (GM) and Gray-Scott (GS) models, we analyze three examples of delay resulting from slow passage into regimes of oscillatory and competition instability. In the first example, for the GM model on the infinite real line, we analyze the delay resulting from slowly tuning a control parameter through a Hopf bifurcation. In the second example, we consider a Hopf bifurcation on a finite one-dimensional domain. In this scenario, as opposed to the extrinsic tuning of a system parameter through a bifurcation value, we analyze the delay of a bifurcation triggered by slow intrinsic dynamics of the PDE system. In the third example, we consider competition instabilities of the GS model triggered by the extrinsic tuning of a feed rate parameter. In all cases, we find that the system must pass well into the unstable regime before the onset of instability is fully observed, indicating delay. We also find that delay has an important effect on the eventual dynamics of the system in the unstable regime. We give analytic predictions for the magnitude of the delays as obtained through analysis of certain explicitly solvable nonlocal eigenvalue problems. The theory is confirmed by numerical solutions of the full PDE systems.Comment: 31 pages, 20 figures, submitted to Physica D: Nonlinear Phenomen

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

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