Random replicators with asymmetric couplings

TL;DR

This paper investigates the dynamics of asymmetric random replicator systems using generating functional techniques, revealing phase transitions and species survival regimes, supported by numerical simulations.

Contribution

It introduces a dynamical approach to analyze asymmetric couplings in replicator models, extending beyond symmetric cases and deriving effective single-species descriptions.

Findings

Asymmetry affects order parameters and phase behavior.

Different phase transitions are identified and related.

Finite species survival regimes are observed in cubic interactions.

Abstract

Systems of interacting random replicators are studied using generating functional techniques. While replica analyses of such models are limited to systems with symmetric couplings, dynamical approaches as presented here allow specifically to address cases with asymmetric interactions where there is no Lyapunov function governing the dynamics. We here focus on replicator models with Gaussian couplings of general symmetry between p>=2 species, and discuss how an effective description of the dynamics can be derived in terms of a single-species process. Upon making a fixed point ansatz persistent order parameters in the ergodic stationary states can be extracted from this process, and different types of phase transitions can be identified and related to each other. We discuss the effects of asymmetry in the couplings on the order parameters and the phase behaviour for p=2 and p=3. Numerical simulations verify our theory. For the case of cubic interactions numerical experiments indicate regimes in which only a finite number of species survives, even when the thermodynamic limit is considered.