Analytical approach to bit-string models of language evolution

A formulation of bit-string models of language evolution, based on differential equations for the population speaking each language, is introduced and preliminarily studied. Connections with replicator dynamics and diffusion processes are pointed out. The stability of the dominance state, where most of the population speaks a single language, is analyzed within a mean-field-like approximation, while the homogeneous state, where the population is evenly distributed among languages, can be exactly studied. This analysis discloses the existence of a bistability region, where dominance coexists with homogeneity as possible asymptotic states. Numerical resolution of the differential system validates these findings.

💡 Research Summary

The paper introduces a continuous‑time, differential‑equation framework for the class of “bit‑string” language‑evolution models, where each possible language is represented by a binary string of length L. In this representation there are 2^L distinct languages, and the state of the system at any time is given by the population numbers n_i(t) for each language i. The authors formulate the dynamics as a combination of two elementary processes. First, a diffusion‑like mutation term allows speakers to switch from language i to language j at a rate p_{ij} that is proportional to the Hamming distance d_{ij} between the two strings; this captures random bit‑flips and thus the creation of new linguistic variants. Second, a replicator term models the selective advantage of a language: the growth rate of language i is taken to be f(x_i) where x_i = n_i/N is the fraction of the total population speaking that language and f is an increasing function (often taken linear, f(x)=αx+β). The full set of equations reads

\