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 s tationary 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 s elf-replicating pulses. For each solution constructed in A. Doelman et al. [Nonlinearity 10 (1997) 523-563], we analytically find a large op en region in the space of the two scaled parameters in which it is sta ble. Specifically, for each value of the scaled inhibitor feed rate, t here exists an interval, whose length and location depend on the solut ion type, of values of the activator (autocatalyst) decay rate for whi ch the solution is stable. The upper boundary of each interval corresp onds to a subcritical Hopf bifurcation point, and the lower boundary i s explicitly determined by finding the parameter value where the solut ion '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 par ameter is decreased, and then successively, the spatially periodic sol utions (with decreasing period) become stable. Moreover, the stability intervals for different solutions overlap. These stability results ar e derived via the reduction of a fourth-order slow-fast eigenvalue pro blem to a second-order nonlocal eigenvalue problem (NLEP), Explicit de termination of these stability intervals plays a central role in under standing pulse self-replication. Numerical simulations confirm that th e spatially periodic stationary solutions are attractors in the pulse- splitting regime; and, moreover, whenever, for a given solution, the v alue of the activator decay rate was taken to lie in the regime below that solution's stability interval, initial data close to that solutio n were observed to evolve toward a different spatially periodic statio nary solution, one whose stability interval included the parameter val ue. The main analytical technique used is that of matched asymptotic e xpansions. (C) 1998 Elsevier Science B.V.