The Stability of Localized Spikes for the 1-D Brusselator Reaction-Diffusion Model

In a one-dimensional domain, the stability of localized spike patterns is analyzed for two closely related singularly perturbed reaction-diffusion (RD) systems with Brusselator kinetics. For the first system, where there is no influx of the inhibitor on the domain boundary, asymptotic analysis is used to derive a nonlocal eigenvalue problem (NLEP) whose spectrum determines the linear stability of a multi-spike steady-state solution. Similar to previous NLEP stability analyses of spike patterns for other RD systems, such as the Gierer-Meinhardt (GM) and Gray-Scott (GS) models, a multi-spike steady-state solution can become unstable to either a competition or an oscillatory instability depending on the parameter regime. An explicit result for the threshold value for the initiation of a competition instability, which triggers the annihilation of spikes in a multi-spike pattern, is derived. Alternatively, in the parameter regime when a Hopf bifurcation occurs, it is shown from a numerical study of the NLEP that an asynchronous, rather than synchronous, oscillatory instability of the spike amplitudes can be the dominant instability. The existence of robust asynchronous temporal oscillations of the spike amplitudes has not been predicted from NLEP stability studies of other RD systems. For the second system, where there is an influx of inhibitor from the domain boundaries, an NLEP stability analysis of a quasi-steady-state two-spike pattern reveals the possibility of dynamic bifurcations leading to either a competition or an oscillatory instability of the spike amplitudes depending on the parameter regime. It is shown that the novel asynchronous oscillatory instability mode can again be the dominant instability. For both Brusselator systems, the detailed stability results from NLEP theory are confirmed by rather extensive numerical computations of the full PDE system.