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Existence and stability of multiple spot solutions for the gray-scott model in R^2
We study the Gray-Scott model in a bounded two dimensional domain and establish the existence and stability of {\bf symmetric} and {\bf asymmetric} multiple spotty patterns. The Green's function and its derivatives
together with two nonlocal eigenvalue problems
both play a major role in the analysis.
For symmetric spots, we establish a threshold behavior for stability:
If a certain inequality for the parameters holds
then we get stability, otherwise we get instability of multiple spot solutions.
For asymmetric spots, we show that they can be stable within a narrow parameter range
Mesa-type patterns in the one-dimensional Brusselator and their stability
The Brusselator is a generic reaction-diffusion model for a tri-molecular
chemical reaction. We consider the case when the input and output reactions are
slow. In this limit, we show the existence of -periodic, spatially bi-stable
structures, \emph{mesas}, and study their stability. Using singular
perturbation techniques, we find a threshold for the stability of mesas.
This threshold occurs in the regime where the exponentially small tails of the
localized structures start to interact. By comparing our results with Turing
analysis, we show that in the generic case, a Turing instability is followed by
a slow coarsening process whereby logarithmically many mesas are annihilated
before the system reaches a steady equilibrium state. We also study a
``breather''-type instability of a mesa, which occurs due to a Hopf
bifurcation. Full numerical simulations are shown to confirm the analytical
results.Comment: to appear, Physica
Stationary Multiple Spots for Reaction-Diffusion Systems
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 reaction-diffusion
systems: activator-inhibitor systems (such as
the Gierer-Meinhardt system) and
activator-substrate 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
Liapunov-Schmidt 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
General theory of instabilities for patterns with sharp interfaces in reaction-diffusion systems
An asymptotic method for finding instabilities of arbitrary -dimensional
large-amplitude patterns in a wide class of reaction-diffusion systems is
presented. The complete stability analysis of 2- and 3-dimensional localized
patterns is carried out. It is shown that in the considered class of systems
the criteria for different types of instabilities are universal. The specific
nonlinearities enter the criteria only via three numerical constants of order
one. The performed analysis explains the self-organization scenarios observed
in the recent experiments and numerical simulations of some concrete
reaction-diffusion systems.Comment: 21 pages (RevTeX), 8 figures (Postscript). To appear in Phys. Rev. E
(April 1st, 1996
Parametric pattern selection in a reaction-diffusion model
We compare spot patterns generated by Turing mechanisms with those generated by replication cascades, in a model one-dimensional reaction-diffusion system. We determine the stability region of spot solutions in parameter space as a function of a natural control parameter (feed-rate) where degenerate patterns with different numbers of spots coexist for a fixed feed-rate. While it is possible to generate identical patterns via both mechanisms, we show that replication cascades lead to a wider choice of pattern profiles that can be selected through a tuning of the feed-rate, exploiting hysteresis and directionality effects of the different pattern pathways