The generalized Gierer-Meinhardt equation is a paradigm model of two coupled reaction-diffusion equations in the theory of biological pattern formation. The existence of time independent, spatially periodic patterns of this equation in one space dimension is governed by a four-dimensional singularly perturbed ordinary differential equation (ODE). In this paper, we analyze this ODE and use geometric singular perturbation theory to construct countably many different oneparameter families of periodic solutions. These periodic orbits consist of fast segments interspersed with slow segments, such that during the fast segments the concentration of one species exhibits finite sequences of large amplitude pulses that occur in rapid succession and that are of one of two fundamental types while the concentration of the other species is constant. Moreover, during the slow segments, the first species is near equilibrium while the concentration of the second decreases slowly. Most significantly, these fast-slow periodic orbits are shown to exist for all periodic sequences of pulses with two reflection symmetries. Hence, the Gierer-Meinhardt model possesses a rich set of stationary, spatially periodic, multi-pulse patterns. We also treat the case of bounded intervals with Neumann boundary conditions. We show that, given any random sequence of the two pulse types, there exists a stationary, in general aperiodic, multi-pulse pattern with the pulses ordered precisely as in the random sequence on sufficiently long intervals.
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Doelman et al. (2001) studied this question.