Multi-spike solutions of a hybrid reaction-transport model

Numerical 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 $D_a\ll D_h$, where $D_a$ and $D_h$ are the diffusivities of the activator and inhibitor, respectively. Asymptotic analysis can then be used to analyze the existence and linear stability of multi-spike solutions. In this paper, we analyze 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. This class of model was recently introduced to account for the formation and homeostatic regulation of synaptic puncta during larval development in {\em C. elegans}. We exploit the fact that that the hybrid model can be mapped onto the classical GM model in the fast switching limit $\alpha\rightarrow \infty$, which allows us to establish the existence of multi-spike solutions. Linearization about the multi-spike solution leads to a non-local eigenvalue problem that is used to derive stability conditions for the multi-spike solution for finite $\alpha$