Phase Transitions in Systems of Interacting Species

We discuss an autocatalytic reaction system: the cyclic competition model A1 + A2 → 2A2, A2 + A3 → 2A3, A3 + A4 → 2A4, A4 + A1 → 2A1), as well as its neutral counterpart. Migrations are introduced into the model. When stochastic phenomena are taken into account, a phase transition between a “fixation” and a “neutral” regime is observed. In the “fixation” regime, species A1 and A3 form an alliance against species A2 and A4, and the final state is one in which one of the symbiotic pairs has won. The odd–even “coarse–grained” systems is mapped onto the two–species neutral (Kimura) model. In the “neutral” regime, all four species survive for long (evolutionary) times. The analytical results are checked against computer simulations of the model. The model is generalized for n species. Also, a generalized version of the Volterra model is analysed.