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The emergence of a spatially-organized population distribution depends on thedynamics of the population and mediators of interaction (activators andinhibitors). Two broad classes of models have been used to investigate when andhow self-organization is triggered, namely, reaction-diffusion and spatiallynonlocal models. Nevertheless, these models implicitly assume smoothpropagation scenarios, neglecting that individuals many times interact byexchanging short and abrupt pulses of the mediating substance. A recentlyproposed framework advances in the direction of properly accounting for theseshort-scale fluctuations by applying a coarse-graining procedure on the pulsedynamics. In this paper, we generalize the coarse-graining procedure and applythe extended formalism to new scenarios in which mediators influenceindividuals' reproductive success or their motility. We show that, in the slow-and fast-mediator limits, pulsed interactions recover, respectively, thereaction-diffusion and nonlocal models, providing a mechanistic connectionbetween them. Furthermore, at each limit, the spatial stability condition isqualitatively different, leading to a timescale-induced transition wherespatial patterns emerge as mediator dynamics becomes sufficiently fast.

Pulsed interactions unify reaction-diffusion and spatial nonlocal models for biological pattern formation