Relative population size, co-operation pressure and strategy correlation in two-population evolutionary dynamics

We study the coupled dynamics of two populations of random replicators by means of statistical mechanics methods, and focus on the effects of relative population size, strategy correlations and heterogeneities in the respective co-operation pressures. To this end we generalise existing path-integral approaches to replicator systems with random asymmetric couplings. This technique allows one to formulate an effective dynamical theory, which is exact in the thermodynamic limit and which can be solve for persistent order parameters in a fixed-point regime regardless of the symmetry of the interactions. The onset of instability can be determined self-consistently. We calculate quantities such as the diversity of the respective populations and their fitnesses in the stationary state, and compare results with data from a numerical integration of the replicator equations

💡 Research Summary

This paper investigates the coupled evolutionary dynamics of two interacting populations of random replicators using statistical‑mechanics techniques. The authors extend the path‑integral formalism, originally developed for single‑population replicator systems, to handle two populations linked by fully asymmetric random couplings. Each population (A and B) contains N_A and N_B species, respectively, and the interaction matrices J_{ij} (A→B) and K_{ji} (B→A) are drawn independently from Gaussian distributions with zero mean and variances scaled as 1/N_B and 1/N_A. This construction captures the essential asymmetry present in many ecological, economic, or social systems where two groups influence each other in a non‑reciprocal way.

The replicator equations are written as

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