Stability analysis of singular patterns in the 1D Gray-Scott model: a matched asymptotics approach

In this work, we analyze the linear stability of singular homoclinic stationary solutions and spatially periodic stationary solutions in the one-dimensional Gray-Scott model. This stability analysis has several implications for understanding the recently discovered phenomena of self-replicating pulses. For each solution constructed in A. Doelman et al. [Nonlinearity 10 (1997) 523-563], we analytically find a large open region in the space of the two scaled parameters in which it is stable. Specifically, for each value of the scaled inhibitor feed rate, there exists an interval, whose length and location depend on the solution type, of values of the activator (autocatalys0 decay rate for which the solution is stable. The upper boundary of each interval corresponds to a subcritical Hopf bifurcation point, and the lower boundary is explicitly determined by finding the parameter value where the solution 'disappears,' i.e., below which it no longer exists as a solution of the steady state system. Explicit asymptotic formulae show that the one-pulse homoclinic solution gains stability first as the second parameter is decreased, and then successively, the spatially periodic solutions (with decreasing period) become stable. Moreover, the stability intervals for different solutions overlap. These stability results are derived via the reduction of a fourth-order slow-fast eigenvalue problem to a second-order nonlocal eigenvalue problem (NLEP). Explicit determination of these stability intervals plays a central role in understanding pulse self-replication. Numerical simulations confirm that the spatially periodic stationary solutions are attractors in the pulse-splitting regime; and, moreover, whenever, for a given solution, the value of the activator decay rate was taken to lie in the regime below that solution's stability interval, initial data close to that solution were observed to evolve toward a different spatially periodic stationary solution, one whose stability interval included the parameter value. The main analytical technique used is that of matched asymptotic expansions. © 1998 Elsevier Science B.V. PACS: 82.40.Ck; 82.40.Bj; 0.2.30.Jr