Localized Patterns in the Gray-Scott Model An Asymptotic and Numerical Study of Dynamics and 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 that occurs in the semi-strong interaction regime where the diffusivity ratio of the two solution components is asymptotically small. In a one-dimensional spatial domain, two distinct types of oscillatory instabilities of multi-spike solutions to the Gray-Scott model that occur in different parameter regimes are analyzed. These two instabilities relate to either an oscillatory instability in the amplitudes of the spikes, or an oscillatory instability in the spatial locations of the spikes. In the latter case a novel Stefan-type problem, with moving Dirac source terms, is shown to characterize the dynamics of a collection of spikes. From a numerical and analytical study of this problem, it is shown that an oscillatory motion in the spike locations can be initiated through a Hopf bifurcation. In a subregime of the parameters it is shown that this Stefan-type problem is quasi-steady, allowing for the derivation of an explicit set of ODE’s for the spike dynamics. In this subregime, a nonlocal eigenvalue problem analysis shows that spike amplitude oscillations can occur from another Hopf bifurcation. In a two-dimensional domain, the method of matched asymptotic expansions is used to construct multi-spot solutions by effectively summing an infinite-order logarithmic expansion in terms of a small parameter. An asymptotic differential algebraic system of ODE’s for the spot locations is derived to characterize the slow dynamics of a collection of spots. Furthermore, it is shown theoretically and from the numerical computation of certain eigenvalue problems that there are three main types of fast instabilities for a multi-spot solution. These instabilities are spot self-replication, spot annihilation due to overcrowding, and an oscillatory instability in the spot amplitudes. These instability mechanisms are studied in detail and phase diagrams in parameter space where they occur are computed and illustrated for various spatial configurations of spots and several domain geometries.