  How large ecosystems can create and maintain the remarkable biodiversity we
see in nature is probably one of the biggest open question in science,
attracting attention from different fields, from Theoretical Ecology to
Mathematics and Physics. In this context, modeling the stable coexistence of
different species competing for limited resources is a particularly demanding
task. From the mathematical point of view, coexistence in competitive dynamics
can be achieved when dominance among species forms intransitive loops. However,
these relationships usually lead to species' densities neutrally cycling
without converging to a stable equilibrium and, although in recent years
several mechanisms have been proposed, models able to explain the robust
persistence of competitive ecosystems are lacking. Here we show that stable
coexistence in large communities can be achieved when the locality of
interactions is taken into account. We consider a simplified ecosystem where
individuals of each species lay on a spatial network and interactions are
possible only between nodes at a certain distance. Varying such distance allows
to interpolate between local and global competition. Our results demonstrate
that when two conditions are met: individuals are embedded in space and can
only interact with other individuals within a short distance, species coexist
reaching a stable equilibrium. On the contrary, when one of these ingredients
is missing large fluctuations and neutral cycles emerge.
