Mean-Field and Measure-Valued Differential Equation Models for Language Variation and Change in a Spatially Distributed Population

Abstract

Although discrete formalisms have been successful at describing the sets of grammatical sentences in human languages, new tools are needed to model language variation. An individual's speech pattern can be modeled more realistically by a stochastic grammar consisting of a set of idealized grammars together with a set of usage rates. A population can then be represented as a probability measure over a space of usage rates and physical or social locations. In this article, we investigate a measure-valued differential equation for a spatially distributed population in which individuals use stochastic grammars. Under appropriate hypotheses, and assuming that children learn based on an average feature of the nearby population's speech, the asymptotic behavior of the measure dynamics is controlled by the feature's dynamics, which can significantly reduce the dimension of the model. We discuss the example of a single usage rate for choosing one of two grammatical options. If space is unstructured, then all populations tend to a stable equilibrium dominated by one option or the other. If space consists of two well-mixed compartments, then each compartment may choose a different dominant idealized grammar, but increased migration causes a bifurcation in which one idealized grammar goes extinct. If space is continuous, numerical experiments show that the measure and feature dynamics can exhibit traveling waves.

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