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We present a non-equilibrium statistical mechanics description of rankabundance relations (RAR) in random community models of ecology. Specifically,we study a multi-species replicator system with quenched random interactionmatrices. We here consider symmetric interactions as well as asymmetric andanti-symmetric cases. RARs are obtained analytically via a generatingfunctional analysis, describing fixed-point states of the system in terms of asmall set of order parameters, and in dependence on the symmetry or otherwiseof interactions and on the productivity of the community. Our work is anextension of Tokita [Phys. Rev. Lett. {\bf 93} 178102 (2004)], where the caseof symmetric interactions was considered within an equilibrium setup. Thespecies abundance distribution in our model come out as truncated normaldistributions or transformations thereof and, in some case, are similar toleft-skewed distributions observed in ecology. We also discuss the interactionstructure of the resulting food-web of stable species at stationarity, cases ofheterogeneous co-operation pressures as well as effects of finite system sizeand of higher-order interactions.

Rank abundance relations in evolutionary dynamics of random replicators