  The emergence of a spatially-organized population distribution depends on the
dynamics of the population and mediators of interaction (activators and
inhibitors). Two broad classes of models have been used to investigate when and
how self-organization is triggered, namely, reaction-diffusion and spatially
nonlocal models. Nevertheless, these models implicitly assume smooth
propagation scenarios, neglecting that individuals many times interact by
exchanging short and abrupt pulses of the mediating substance. A recently
proposed framework advances in the direction of properly accounting for these
short-scale fluctuations by applying a coarse-graining procedure on the pulse
dynamics. In this paper, we generalize the coarse-graining procedure and apply
the extended formalism to new scenarios in which mediators influence
individuals' reproductive success or their motility. We show that, in the slow-
and fast-mediator limits, pulsed interactions recover, respectively, the
reaction-diffusion and nonlocal models, providing a mechanistic connection
between them. Furthermore, at each limit, the spatial stability condition is
qualitatively different, leading to a timescale-induced transition where
spatial patterns emerge as mediator dynamics becomes sufficiently fast.
