8,330 research outputs found
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
Stationary localized structures and the effect of the delayed feedback in the Brusselator model
The Brusselator reaction-diffusion model is a paradigm for the understanding
of dissipative structures in systems out of equilibrium. In the first part of
this paper, we investigate the formation of stationary localized structures in
the Brusselator model. By using numerical continuation methods in two spatial
dimensions, we establish a bifurcation diagram showing the emergence of
localized spots. We characterize the transition from a single spot to an
extended pattern in the form of squares. In the second part, we incorporate
delayed feedback control and show that delayed feedback can induce a
spontaneous motion of both localized and periodic dissipative structures. We
characterize this motion by estimating the threshold and the velocity of the
moving dissipative structures.Comment: 18 pages, 11 figure
Stripe to spot transition in a plant root hair initiation model
A generalised Schnakenberg reaction-diffusion system with source and loss
terms and a spatially dependent coefficient of the nonlinear term is studied
both numerically and analytically in two spatial dimensions. The system has
been proposed as a model of hair initiation in the epidermal cells of plant
roots. Specifically the model captures the kinetics of a small G-protein ROP,
which can occur in active and inactive forms, and whose activation is believed
to be mediated by a gradient of the plant hormone auxin. Here the model is made
more realistic with the inclusion of a transverse co-ordinate. Localised
stripe-like solutions of active ROP occur for high enough total auxin
concentration and lie on a complex bifurcation diagram of single and
multi-pulse solutions. Transverse stability computations, confirmed by
numerical simulation show that, apart from a boundary stripe, these 1D
solutions typically undergo a transverse instability into spots. The spots so
formed typically drift and undergo secondary instabilities such as spot
replication. A novel 2D numerical continuation analysis is performed that shows
the various stable hybrid spot-like states can coexist. The parameter values
studied lead to a natural singularly perturbed, so-called semi-strong
interaction regime. This scaling enables an analytical explanation of the
initial instability, by describing the dispersion relation of a certain
non-local eigenvalue problem. The analytical results are found to agree
favourably with the numerics. Possible biological implications of the results
are discussed.Comment: 28 pages, 44 figure
Adiabatic stability under semi-strong interactions: The weakly damped regime
We rigorously derive multi-pulse interaction laws for the semi-strong
interactions in a family of singularly-perturbed and weakly-damped
reaction-diffusion systems in one space dimension. Most significantly, we show
the existence of a manifold of quasi-steady N-pulse solutions and identify a
"normal-hyperbolicity" condition which balances the asymptotic weakness of the
linear damping against the algebraic evolution rate of the multi-pulses. Our
main result is the adiabatic stability of the manifolds subject to this normal
hyperbolicity condition. More specifically, the spectrum of the linearization
about a fixed N-pulse configuration contains essential spectrum that is
asymptotically close to the origin as well as semi-strong eigenvalues which
move at leading order as the pulse positions evolve. We characterize the
semi-strong eigenvalues in terms of the spectrum of an explicit N by N matrix,
and rigorously bound the error between the N-pulse manifold and the evolution
of the full system, in a polynomially weighted space, so long as the
semi-strong spectrum remains strictly in the left-half complex plane, and the
essential spectrum is not too close to the origin
Oscillatory translational instabilities of localized spot patterns in the Schnakenberg reaction-diffusion system on general 2-D domains
For a bounded 2-D planar domain , we investigate the impact of domain
geometry on oscillatory translational instabilities of -spot equilibrium
solutions for a singularly perturbed Schnakenberg reaction-diffusion system
with \mO(\eps^2) \ll \mO(1) activator-inhibitor diffusivity ratio. An
-spot equilibrium is characterized by an activator concentration that is
exponentially small everywhere in except in well-separated
localized regions of \mO(\eps) extent. We use the method of matched
asymptotic analysis to analyze Hopf bifurcation thresholds above which the
equilibrium becomes unstable to translational perturbations, which result in
\mO(\eps^2)-frequency oscillations in the locations of the spots. We find
that stability to these perturbations is governed by a nonlinear
matrix-eigenvalue problem, the eigenvector of which is a -vector that
characterizes the possible modes (directions) of oscillation. The
matrix contains terms associated with a certain Green's function on ,
which encodes geometric effects. For the special case of a perturbed disk with
radius in polar coordinates with \red{}, , and
-periodic, we show that only the mode- coefficients of the Fourier
series of impact the bifurcation threshold at leading order in . We
further show that when , the dominant mode of
oscillation is in the direction parallel to the longer axis of the perturbed
disk. Numerical investigations on the full Schnakenberg PDE are performed for
various domains and -spot equilibria to confirm asymptotic results
and also to demonstrate how domain geometry impacts thresholds and dominant
oscillation modes
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