5,503 research outputs found
Instability of Turing patterns in reaction-diffusion-ODE systems
The aim of this paper is to contribute to the understanding of the pattern
formation phenomenon in reaction-diffusion equations coupled with ordinary
differential equations. Such systems of equations arise, for example, from
modeling of interactions between cellular processes such as cell growth,
differentiation or transformation and diffusing signaling factors. We focus on
stability analysis of solutions of a prototype model consisting of a single
reaction-diffusion equation coupled to an ordinary differential equation. We
show that such systems are very different from classical reaction-diffusion
models. They exhibit diffusion-driven instability (Turing instability) under a
condition of autocatalysis of non-diffusing component. However, the same
mechanism which destabilizes constant solutions of such models, destabilizes
also all continuous spatially heterogeneous stationary solutions, and
consequently, there exist no stable Turing patterns in such
reaction-diffusion-ODE systems. We provide a rigorous result on the nonlinear
instability, which involves the analysis of a continuous spectrum of a linear
operator induced by the lack of diffusion in the destabilizing equation. These
results are extended to discontinuous patterns for a class of nonlinearities.Comment: This is a new version of the paper. Presentation of results was
essentially revised according to referee suggestion
Stability analysis and simulations of coupled bulk-surface reaction–diffusion systems
In this article, we formulate new models for coupled systems of bulk-surface reaction–diffusion equations on stationary volumes. The bulk reaction–diffusion equations are coupled to the surface reaction–diffusion equations through linear Robin-type boundary conditions. We then state and prove the necessary conditions for diffusion-driven instability for the coupled system. Owing to the nature of the coupling between bulk and surface dynamics, we are able to decouple the stability analysis of the bulk and surface dynamics. Under a suitable choice of model parameter values, the bulk reaction–diffusion system can induce patterning on the surface independent of whether the surface reaction–diffusion system produces or not, patterning. On the other hand, the surface reaction–diffusion system cannot generate patterns everywhere in the bulk in the absence of patterning from the bulk reaction–diffusion system. For this case, patterns can be induced only in regions close to the surface membrane. Various numerical experiments are presented to support our theoretical findings. Our most revealing numerical result is that, Robin-type boundary conditions seem to introduce a boundary layer coupling the bulk and surface dynamics
Role of hydrodynamic flows in chemically driven droplet division
We study the hydrodynamics and shape changes of chemically active droplets.
In non-spherical droplets, surface tension generates hydrodynamic flows that
drive liquid droplets into a spherical shape. Here we show that spherical
droplets that are maintained away from thermodynamic equilibrium by chemical
reactions may not remain spherical but can undergo a shape instability which
can lead to spontaneous droplet division. In this case chemical activity acts
against surface tension and tension-induced hydrodynamic flows. By combining
low Reynolds-number hydrodynamics with phase separation dynamics and chemical
reaction kinetics we determine stability diagrams of spherical droplets as a
function of dimensionless viscosity and reaction parameters. We determine
concentration and flow fields inside and outside the droplets during shape
changes and division. Our work shows that hydrodynamic flows tends to stabilize
spherical shapes but that droplet division occurs for sufficiently strong
chemical driving, sufficiently large droplet viscosity or sufficiently small
surface tension. Active droplets could provide simple models for prebiotic
protocells that are able to proliferate. Our work captures the key
hydrodynamics of droplet division that could be observable in chemically active
colloidal droplets
Nonlinear stability of stationary solutions for curvature flow with triple junction
In this paper we analyze the motion of a network of three planar curves with
a speed proportional to the curvature of the arcs, having perpendicular
intersections with the outer boundary and a common intersection at a triple
junction. As a main result we show that a linear stability criterion due to
Ikota and Yanagida is also sufficient for nonlinear stability. We also prove
local and global existence of classical smooth solutions as well as various
energy estimates. Finally, we prove exponential stabilization of an evolving
network starting from the vicinity of a linearly stable stationary network.Comment: submitte
Morphological stability of electromigration-driven vacancy islands
The electromigration-induced shape evolution of two-dimensional vacancy
islands on a crystal surface is studied using a continuum approach. We consider
the regime where mass transport is restricted to terrace diffusion in the
interior of the island. In the limit of fast attachment/detachment kinetics a
circle translating at constant velocity is a stationary solution of the
problem. In contrast to earlier work [O. Pierre-Louis and T.L. Einstein, Phys.
Rev. B 62, 13697 (2000)] we show that the circular solution remains linearly
stable for arbitrarily large driving forces. The numerical solution of the full
nonlinear problem nevertheless reveals a fingering instability at the trailing
end of the island, which develops from finite amplitude perturbations and
eventually leads to pinch-off. Relaxing the condition of instantaneous
attachment/detachment kinetics, we obtain non-circular elongated stationary
shapes in an analytic approximation which compares favorably to the full
numerical solution.Comment: 12 page
Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation
In this paper we consider the spectral and nonlinear stability of periodic
traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In
particular, we resolve the long-standing question of nonlinear modulational
stability by demonstrating that spectrally stable waves are nonlinearly stable
when subject to small localized (integrable) perturbations. Our analysis is
based upon detailed estimates of the linearized solution operator, which are
complicated by the fact that the (necessarily essential) spectrum of the
associated linearization intersects the imaginary axis at the origin. We carry
out a numerical Evans function study of the spectral problem and find bands of
spectrally stable periodic traveling waves, in close agreement with previous
numerical studies of Frisch-She-Thual, Bar-Nepomnyashchy,
Chang-Demekhin-Kopelevich, and others carried out by other techniques. We also
compare predictions of the associated Whitham modulation equations, which
formally describe the dynamics of weak large scale perturbations of a periodic
wave train, with numerical time evolution studies, demonstrating their
effectiveness at a practical level. For the reader's convenience, we include in
an appendix the corresponding treatment of the Swift-Hohenberg equation, a
nonconservative counterpart of the generalized Kuramoto-Sivashinsky equation
for which the nonlinear stability analysis is considerably simpler, together
with numerical Evans function analyses extending spectral stability analyses of
Mielke and Schneider.Comment: 78 pages, 11 figure
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