Multi-spike solutions of a hybrid reaction-transport model

Simulations of classical pattern-forming reaction-diffusion systems indicate that they often operate in the strongly nonlinear regime, with the final steady state consisting of a spatially repeating pattern of localized spikes. In activator-inhibitor systems such as the two-component Gierer-Meinhardt (GM) model, one can consider the singular limit D-a << D-h, where D-a and D-h are the diffusivities of the activator and inhibitor, respectively. Asymptotic analysis can then be used to analyse the existence and linear stability of multi-spike solutions. In this paper, we analyse multi-spike solutions in a hybrid reaction-transport model, consisting of a slowly diffusing activator and an actively transported inhibitor that switches at a rate alpha between right-moving and left-moving velocity states. Such a model was recently introduced to account for the formation and homeostatic regulation of synaptic puncta during larval development in Caenorhabditis elegans. We exploit the fact that the hybrid model can be mapped onto the classical GM model in the fast switching limit alpha -> infinity, which establishes the existence of multi-spike solutions. Linearization about the multi-spike solution yields a non-local eigenvalue problem that is used to investigate stability of the multi-spike solution by combining analytical results for alpha -> infinity with a graphical construction for finite alpha.