Pattern Formation in Reaction-diffusion Models Far from the Turing Regime

In this thesis we analyse three different reaction-diffusion models These are: the Gray-Scott model of an irreversable chemical reaction, the Gierer-Meinhardt model for seashell patters, and the Haus model of mode-locked lasers. In the limit of small diffusivity, all three models exhibit localised spatial patterns. In one dimension, the equilibrium state typically concentrates on a discrete number of points. In two dimensions, the solution may consist of stripes, spots, domain-filling curves, or any combination of these. We study the regime where such structures are very far from the spatially homogenous solution. As such, the classical Turing analysis of small perturbations of homogenous state is not applicable. Instead, we study perturbations from the localised spike-type solutions. In one dimension, the following instbailities are analysed: an overcrowding instability, whereby some of the spikes are annihilated if the initial state contains too many spikes; undercrowding (or splitting) instability, whereby a new spike may appear by the process of splitting of a spike into two; an oscillatory height instability whereby the spike height oscillates with period of O(1) in time; and an oscillatory drift instability where the center of the spike exhibits a slow, periodic motion. Explicit thresholds on the parameters are derived for each type of instability. In two dimensions, we study spike, stripe and ring-like solutions. For stripe and ring-like solutions, the following instabilities are analysed: a splitting instability, whereby a stripe selfreplicates into two parallel stripes; a breakup instability, where a stripe breaks up into spots; and a zigzag instability, whereby a stripe develops a wavy pattern in the transveral direction. For certain parameter ranges, we derive explicit instability thresholds for all three types of instability. Numerical simulations are used to confirm our analytical predictions. Further numerical simulations are performed, suggesting the existence of a regime where a stripe is stable with respect to breakup or splitting instabilities, but unstable with respect to zigzag instabilities. Based on numerics, we speculate that this leads to domain-filling patterns, and labyrinth-like patterns. For a single spike in two dimensions, we derive an ODE that governs the slow drift of its center. We reduce this problem to the study of the properties of a certain Green’s function. For a specific dumbell-like domain, we obtain explicit formulas for such a Green’s function using complex analysis. This in turn leads to conjecture that under certain general conditions on parameters, the equilibrium location of the spike is unique, for an arbitrary shaped domain. Finally, we consider another parameter regime, for which the exponentially weak interaction of the spike with the boundary plays a crucial role. We show that in this case there can exist a spike equilibrium solution that is located very near the boundary. Such solution is found to be unstable in the direction that is transversal to the boundary. As the effect of the boundary is increased, the interior spike locations undergo a series of destabilizing bifurcations, until all interior spike equilibria become unstable.