  In order to model real ecological systems one has to consider many species
that interact in complex ways. However, most of the recent theoretical studies
have been restricted to few species systems with rather trivial interactions.
The few studies dealing with larger number of species and/or more complex
interaction schemes are mostly restricted to numerical explorations. In this
paper we determine, starting from the deterministic mean-field rate equations,
for large classes of systems the space of coexistence fixed points at which
biodiversity is maximal. For systems with a single coexistence fixed point we
derive complex Ginzburg-Landau equations that allow to describe space-time
pattern realized in two space dimensions. For selected cases we compare the
theoretical predictions with the pattern observed in numerical simulations.
