  How large ecosystems can create and maintain the remarkable biodiversity we
see in nature is probably one of the biggest open questions in science,
attracting attention from different fields, from Theoretical Ecology to
Mathematics and Physics. In this context, modeling the stable coexistence of
species competing for limited resources is a particularly challenging task.
From a 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' relative abundances neutrally cycling
without converging to a stable equilibrium. Although in recent years several
mechanisms have been proposed, models able to explain species coexistence in
competitive communities are still limited. Here we identify locality in the
interactions as one of the simplest mechanisms leading to stable species
coexistence. We consider a simplified ecosystem where individuals of each
species lay on a spatial network and interactions are possible only between
nodes within a certain distance. Varying such distance allows to interpolate
between local and global competition. Our results demonstrate, within the scope
of our model, that species coexist reaching a stable equilibrium when two
conditions are met: individuals are embedded in space and can only interact
with other individuals within a short distance. On the contrary, when one of
these ingredients is missing, large oscillations and neutral cycles emerge.
