In order to understand the role of space in ecological communities where each
species produces a certain type of resource and has varying abilities to
exploit the resources produced by its own species and by the other species, we
carry out a comparative study of an interacting particle system and its
mean-field approximation. For a wide range of parameter values, we show both
analytically and numerically that the spatial model results in predictions that
significantly differ from its nonspatial counterpart, indicating that the use
of the mean-field approach to describe the evolution of communities in which
individuals only interact locally is invalid. In two-species communities, the
disagreements between the models appear either when both species compete by
producing resources that are more beneficial for their own species or when both
species cooperate by producing resources that are more beneficial for the other
species. In particular, while both species coexist if and only if they
cooperate in the mean-field approximation, the inclusion of space in the form
of local interactions may prevent coexistence even in cooperative communities.
Introducing additional species, cooperation is no longer the only mechanism
that promotes coexistence. We prove that, in three-species communities,
coexistence results either from a global cooperative behavior, or from
rock-paper-scissors type interactions, or from a mixture of these dynamics,
which excludes in particular all cases in which two species compete.