Replicators with random interactions: A solvable model

Mean-field methods of statistical physics are applied to replicator selection with N randomly interacting species and deterministic self-interaction u. We present the outcome of a replica symmetric calculation for the global maximum of the mean fitness which is a Lyapunov function of the system. The replica symmetric solution is stable and no further equilibrium states exist at least for u> {}2 . On the other hand, in the low-u region the dynamics relaxes to states which differ from the replica symmetric result.