Self-replicating spots in reaction-diffusion systems

Winter

2007

J. Math. Biol.

Abstract. In this paper, we review analytical methods for a rigorous study of the existence and stability of stationary, multiple spots for reaction-diffusion systems. We will consider two classes of reactiondiffusion systems: activator-inhibitor systems (such as the Gierer-Meinhardt system) and activatorsubstrate systems (such as the Gray-Scott system or the Schnakenberg model).The main ideas are presented in the context of the Schnakenberg model, and these results are new to the literature.We will consider the systems in a two-dimensional, bounded and smooth domain for small diffusion constant of the activator.Existence of multi-spots is proved using tools from nonlinear functional analysis such as LiapunovSchmidt reduction and fixed-point theorems. The amplitudes and positions of spots follow from this analysis.Stability is shown in two parts, for eigenvalues of order one and eigenvalues converging to zero, respectively. Eigenvalues of order one are studied by deriving their leading-order asymptotic behavior and reducing the eigenvalue problem to a nonlocal eigenvalue problem (NLEP). A study of the NLEP reveals a condition for the maximal number of stable spots.Eigenvalues converging to zero are investigated using a projection similar to Liapunov-Schmidt reduction and conditions on the positions for stable spots are derived. The Green's function of the Laplacian plays a central role in the analysis.The results are interpreted in the biological, chemical and ecological contexts. They are confirmed by numerical simulations.

Section: Previous Results On Peaked Solutionsmentioning

confidence: 99%

Stationary multiple spots for reaction–diffusion systems

Winter

2007

J. Math. Biol.

“…These numerical and experimental studies have stimulated much theoretical work to classify steady-state and time-dependent spike behavior in the simpler case of one spatial dimension, including: spike-replication and dynamics in the weak-interaction regime (cf. [39], [41], [36], [44]); spatio-temporal chaos in the weak-interaction regime (cf. [37]); the existence and stability of equilibrium solutions in the semi-strong interaction regime (cf.…”

Section: Introductionmentioning

confidence: 99%

Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Kolokolnikov

Ward

2006

Interfaces Free Bound.

Slow translational instabilities of symmetric k-spike equilibria for the one-dimensional singularly perturbed two-component Gray-Scott (GS) model are analyzed. These symmetric spike patterns are characterized by a common value of the spike amplitude. The GS model is studied on a finite interval in the semi-strong spike-interaction regime, where the diffusion coefficient of only one of the two chemical species is asymptotically small. Two distinguished limits for the GS model are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime it is shown analytically that k−1 small eigenvalues, governing the translational stability of a symmetric kspike pattern, simultaneously cross through zero at precisely the same parameter value at which k −1 different asymmetric k-spike equilibria bifurcate off of the symmetric k-spike equilibrium branch. These asymmetric equilibria have the general form SBB . . . BS (neglecting the positioning of the B and S spikes in the overall spike sequence). For a one-spike equilibrium solution in the intermediate regime it is shown that a translational, or drift, instability can occur from a Hopf bifurcation in the spike-layer location when a reaction-time parameter τ is asymptotically large as ε → 0. Locally, this instability leads to small-scale oscillations in the spike-layer location. For a certain parameter range within the intermediate regime such a drift instability for the GS model is shown to be the dominant instability mechanism. Numerical experiments are performed to validate the asymptotic theory.

“…However, in 1-D, self-replication patterns were observed when D U = 1, D V = δ 2 = 0.01. Some formal asymptotics and dynamics in 1-D are contained in [25] and [24]. Recent numerical simulations of [6] in 1-D and [22], [20] in 2-D show that the single spot may be stable in some very narrow parameter regimes.…”

Section: Theorem 21 (Existence Of Asymmetric Solutions) Assume Thamentioning

confidence: 99%

“…In 1-D, numerical simulations were done by Reynolds, Pearson and PonceDawson [25], [26], independently by Petrov, Scott and Showalter [24] and again self-replication phenomena were observed. However, in 1-D, self-replication patterns were observed when D U = 1, D V = δ 2 = 0.01.…”

Section: Theorem 21 (Existence Of Asymmetric Solutions) Assume Thamentioning

confidence: 99%

Asymmetric Spotty Patterns for the Gray–Scott Model in R²

Winter

2002

Stud Appl Math

Abstract. In this paper, we rigorously prove the existence and stability of asymmetric spotty patterns for the Gray-Scott model in a bounded two dimensional domain. We show that given any two positive integers k 1 , k 2 , there are asymmetric solutions with k 1 large spots (type A) and k 2 small spots (type B). We also give conditions for their location and calculate their heights. Most of these asymmetric solutions are shown to be unstable. However, in a narrow range of parameters, asymmetric solutions may be stable.