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Precursor gradients in a reaction-diffusion system are spatially varyingcoefficients in the reaction-kinetics. Such gradients have been used in variousapplications, such as the head formation in the Hydra, to model the effect ofpre-patterns and to localize patterns in various spatial regions. For the 1-DGierer-Meinhardt (GM) model we show that a simple precursor gradient in thedecay rate of the activator can lead to the existence of stable, asymmetric,two-spike patterns, corresponding to localized peaks in the activator ofdifferent heights. This is a qualitatively new phenomena for the GM model, inthat asymmetric spike patterns are all unstable in the absence of the precursorfield. Through a determination of the global bifurcation diagram of two-spikesteady-state patterns, we show that asymmetric patterns emerge from asupercritical symmetry-breaking bifurcation along the symmetric two-spikebranch as a parameter in the precursor field is varied. Through a combinedanalytical-numerical approach we analyze the spectrum of the linearization ofthe GM model around the two-spike steady-state to establish that portions ofthe asymmetric solution branches are linearly stable. In this linear stabilityanalysis a new class of vector-valued nonlocal eigenvalue problem (NLEP) isderived and analyzed.

Stable Asymmetric Spike Equilibria for the Gierer-Meinhardt Model with a Precursor Field