Language competition on lattices

Abstract

AbstractWe consider a language dynamics ODE model for two languages on a square lattice. The model is an extension of the one popularized by Abrams and Strogatz. In our study, 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 adjacent sites is nonlinear, the transition between regions speaking two different languages is super‐exponential. The full dynamics are considered as a function of the prestige of a language. It is seen that as the prestige varies, it allows for a language to spread through the lattice, or conversely for its demise. Although most of the work is done assuming a square lattice, we briefly consider the problem on a triangle lattice to illustrate the robustness of the findings. We conclude by looking at the existence and spectral stability of fronts and pulses for an associated continuum PDE model. In this model, the super‐exponential decay associated with the lattice model is reflected in the existence of compactly supported structures (compactons).