  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 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 is addressed.
