  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 $\alpha$
