Related papers: Stable localized moving patterns in the 2-D Gray-S…
The dynamics and stability of multi-spot patterns to the Gray-Scott (GS) reaction-diffusion model in a two-dimensional domain is studied in the singularly perturbed limit of small diffusivity $\epsilon$ of one of the two solution…
Localized spot patterns, where one or more solution components concentrates at certain points in the domain, are a common class of localized pattern for reaction-diffusion systems, and they arise in a wide range of modeling scenarios. In an…
In two-dimensional space, we investigate the slow dynamics of multiple localized spots with oscillatory tails in a specific three-component reaction-diffusion system, whose key feature is that the spots attract or repel each other…
Localized patterns in singularly perturbed reaction-diffusion equations typically consist of slow parts -- in which the associated solution follows an orbit on a slow manifold in a reduced spatial dynamical system -- alternated by fast…
In this paper, we explore pattern formation in a four-species variational Gary-Scott model, which includes all reverse reactions and introduces a virtual species to describe the birth-death process in the classical Gray-Scott model. This…
The study of pattern emergence together with exploration of the exemplar Turing model is enjoying a renaissance both from theoretical and experimental perspective. Here, we implement a stability analysis of spatially dependent reaction…
We consider the stability of position control of traveling waves in reaction-diffusion system as proposed in {[}J. L\"ober, H. Engel, arXiv:1304.2327{]}. Instead of analyzing the controlled reaction-diffusion system, stability is studied on…
We investigate Turing instability and pattern formation in two-dimensional domains for two reaction-diffusion models, obtained as diffusive limits of kinetic equations for mixtures of monatomic and polyatomic gases. The first model is of…
The existence and stability of localized patterns of criminal activity are studied for the reaction-diffusion model of urban crime that was introduced by Short et. al. [Math. Models. Meth. Appl. Sci., 18, Suppl. (2008), pp. 1249--1267].…
In the singularly perturbed limit corresponding to a large diffusivity ratio between two components in a reaction-diffusion (RD) system, quasi-equilibrium spot patterns are often admitted, producing a solution that concentrates at a…
Reaction-diffusion equation model for a 2-dimensional gas discharge system is introduced in close relationship with the recent experiment by Nasuno. The model shows formation of spots, molecule-like organization of a cluster of spots,…
In this article, we carry out a study of long-term behavior of reaction-diffusion systems augmented with self- and cross-diffusion, using an augmented Gray-Scott system as a general example. The methodology remains generic, and is therefore…
This paper investigates the conditions for the stability and emergence of patterns in a new three-component reaction-diffusion system. The system describes the coexistence and interaction of water reservoirs, vegetation, and bushfire…
An asymptotic method for finding instabilities of arbitrary $d$-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…
The non-equilibrium dynamic fluctuations of a stochastic version of the Gray-Scott (GS) model are studied analytically in leading order in perturbation theory by means of the dynamic renormalization group. There is an attracting stable…
We develop a complete stability theory for two-dimensional periodic traveling waves of reaction-diffusion systems. More precisely, we identify a diffusive spectral stability assumption, prove that it implies nonlinear stability and provide…
The replication and differentiation of spots in reaction diffusion equations are studied by extending the Gray-Scott model with self-replicating spots to include many degrees of freedom needed to model systems with many chemicals. By…
We performed a comprehensive study of the spike autosolitons: self-sustained solitary inhomogeneous states, in the classical reaction-diffusion system --- the Gray-Scott model. We developed singular perturbation techniques based on the…
We consider reaction-diffusion equations on a thick curved surface and obtain a set of effective R-D equation to ${\cal O}(\epsilon^2)$, where $\epsilon$ is the surface thickness. We observe that the R-D systems on these curved surfaces can…
A general system of several ordinary differential equations coupled with a reaction-diffusion equation in a bounded domain with zero-flux boundary condition is studied in the context of pattern formation. These initial-boundary value…