Dynamics of two languages competing on a network: a case study

TL;DR

This paper analyzes a nonlinear lattice model of two competing languages, exploring how language dominance structures like stripes and spots form and change with language prestige, using bifurcation theory.

Contribution

It extends a known language competition model to a lattice framework and studies the stability of complex spatial structures using ODE bifurcation analysis.

Findings

Structures like stripes and spots can form and remain stable.

Language prestige influences the spread or decline of a language.

Compactly supported pulses (compactons) can exist in the model.

Abstract

A language dynamics model on a square lattice, which is an extension of the one popularized by Abrams and Strogatz [1], is analyzed using ODE bifurcation theory. For this model we are interested in the existence and spectral stability of structures such as stripes, which are realized through pulses and/or the concatenation of fronts, and spots, which are a contiguous collection of sites in which one language is dominant. Because the coupling between sites is nonlinear, the boundary between sites containing speaking two different languages is "sharp"; in particular, in a PDE approximation it allows for the existence of compactly supported pulses (compactons). The dynamics are considered as a function of the prestige of a language. In particular, it is seen that as the prestige varies, it allows for a language to spread through the network, or conversely for its demise.