  We present a non-equilibrium statistical mechanics description of rank
abundance relations (RAR) in random community models of ecology. Specifically,
we study a multi-species replicator system with quenched random interaction
matrices. We here consider symmetric interactions as well as asymmetric and
anti-symmetric cases. RARs are obtained analytically via a generating
functional analysis, describing fixed-point states of the system in terms of a
small set of order parameters, and in dependence on the symmetry or otherwise
of interactions and on the productivity of the community. Our work is an
extension of Tokita [Phys. Rev. Lett. {\bf 93} 178102 (2004)], where the case
of symmetric interactions was considered within an equilibrium setup. The
species abundance distribution in our model come out as truncated normal
distributions or transformations thereof and, in some case, are similar to
left-skewed distributions observed in ecology. We also discuss the interaction
structure of the resulting food-web of stable species at stationarity, cases of
heterogeneous co-operation pressures as well as effects of finite system size
and of higher-order interactions.
