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

Abstract

In this work we analyze the linear stability of singular homoclinic stationary solutions and spatiallyperiodic stationary solutions in the onedimensional GrayScott model This stability analysis has several implications for understanding the recently discovered phenomena of selfreplicating pulses For each solution constructed in we analytically nd a large open region in the space of the two scaled parameters in which it is stable Specically 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 autocatalyst 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 nding the parameter value where the solution disappears ie below which it no longer exists as a solution of the steady state system Explicit asymptotic formulae show that the onepulse homoclinic solution gains stability rst as the second parameter is decreased and then successively the spatially periodic solutions with decreasing period become stable Moreover the stability intervals for dierent solutions overlap Explicit determination of these stability intervals plays a central role in understanding pulse selfreplication Numerical simulations conrm that the spatially periodic stationary solutions are attractors in the pulsesplitting regime and moreover whenever for a given solution the value of the activator decay rate was taken to lie in the regime below that solutions stability interval initial data close to that solution was observed to evolve toward a dierent spatially periodic stationary solution one whose stability interval included the parameter value The main analytical technique used is that of matched asymptotic expansions