Two identical 1D autocatalytic systems with Gray–Scott kinetics—driven towards convectively unstable regimes and submitted to independent spatiotemporal Gaussian white noises—are coupled unidirectionally, but otherwise linearly. Numerical simulation then reveals that (even when perturbed by noise) the slave system replicates the convective patterns arising in the master one to a very high degre...

In this paper, we rigorously prove the existence and stability of multiple spot patterns for the Gray-Scott system in a two dimensional domain which are far from spatial homogeneity. The Green's function together with its derivatives and two nonlocal eigenvalue problems both play a major role in the analysis. We establish a threshold behavior: If a certain inequality holds then we get stability...

Localized patterns have been observed in many reaction-diffusion systems. One well-known such system is the two-component Gray-Scott model, which has been shown numerically to exhibit a rich variety of localized spatiotemporal patterns including, standing spots, oscillating spots, self-replicating spots, etc. This thesis concentrates on analyzing the localized pattern formation in this model th...

Two pulse solutions play a central role in the phenomena of self-replicating pulses in the one-dimensional (1-D) Gray–Scott model. In the present work (part I of two parts), we carry out an existence study for solutions consisting of two symmetric pulses moving apart from each other with slowly varying velocities. This corresponds to a “mildly strong” pulse interaction problem in which the inhi...

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 ε of one of the two solution components. A hybrid asymptotic-numerical approach based on combining the method of matched asymptotic expansions with the detailed numerical study of certain eigenvalue proble...

We construct a phenomenological model describing aggregated spots and a loop structure. Our model is based on the Gray-Scott model which is supplemented with a global coupling term and advection terms. One of the species makes a field proportional to its concentration, which induces the advection. By numerically investigating the model, we show that the system has a transition from aggregated s...

Semi-arid ecosystems form the stage for a plethora of vegetation patterns; a feature that has been captured in terms of mathematical models since the beginning of this millennium. To study these patterns, we use a reaction-advection-diffusion model that describes the interaction of vegetation and water supply on gentle slopes. As water diffuses much faster than vegetation, this model operates o...

We consider the upper semicontinuity of the global random attractors for the stochastic three-component reversible Gray–Scott system on unbounded domains when the intensity of the noise converges to zero. 2013 Elsevier Inc. All rights reserved.