On existence and stability of spatial patterns in an activator-inhibitor system exhibiting self-replication

Spontaneous pattern formation is observed in a wide variety of physical, chemical and biological processes. The pioneering work of A. Turing showed that, for reaction-diffusion systems near equilibrium, small amplitude disturbances of unstable homogeneous states can lead to spatially periodic patterns. The patterns form purely as the result of a balance between reaction and diffusion.; This dissertation centers on pattern formation in systems far from equilibrium. We treat the 1-D and 2-D Gray-Scott models, prototypical reaction-diffusion models from chemistry. Novel pattern formation phenomena, including self-replicating pulses, annular rings and spots, are the primary motivations for our interest. We focus on the attractors in the 1-D self-replication regime, and in 2-D, we present one of the first analyses of annular patterns.; In the 1-D Gray-Scott model, we present an analysis of the existence and stability of a complete family of spatially periodic patterns which form a Busse balloon. We show that these patterns are born at a critical parameter value in a Turing/Ginzburg-Landau bifurcation, where their spatial periods are $O$(1). Next, we analytically continue them to the regime where their spatial periods are asymptotically large, using geometric singular perturbation theory and the adiabatic Melnikov method. Depending on parameter values, the family then terminates in global bifurcations or in local bifurcations. Within the existence domain for this family, we also find the Busse balloon in which the stable periodic patterns live. Stability results are obtained using Ginzburg-Landau equation near criticality, and the stability results are numerically continued away from criticality.; In the 2-D Gray-Scott model, we prove the existence of stationary axisymmetric annular patterns via an analytic method. We then perform a stability analysis of these annular solutions, using an extension we formulate of the nonlocal eigenvalue problem (NLEP) method developed by Doelman, Gardner and Kaper. We prove that there exists a band of unstable annular wave-numbers m, and thus we expect an annular solution to break up into spots of a particular wavelength. In simulations of the full 2-D partial differential equation, we find that annular rings do break up into annuli of spots, and that the number of spots on each annulus is correctly predicted by the NLEP method. We also continue these spotted rings into the Turing regime.