Dynamics of random replicators with Hebbian interactions

A system of replicators with Hebbian random couplings is studied using dynamical methods. The self-reproducing species are here characterized by a set of binary traits, and they interact based on complementarity. In the case of an extensive number of traits we use path-integral techniques to demonstrate how the coupled dynamics of the system can be formulated in terms of an effective single-species process in the thermodynamic limit, and how persistent order parameters characterizing the stationary states may be computed from this process in agreement with existing replica studies of the statics. Numerical simulations confirm these results. The analysis of the dynamics allows an interpretation of two different types of phase transitions of the model in terms of memory onset at finite and diverging integrated response, respectively. We extend the analysis to the case of diluted couplings of an arbitrary symmetry, where replica theory is not applicable. Finally the dynamics, and in particular the approach to the stationary state of the model with a finite number of traits, are addressed.