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We provide a unified mathematical explanation of two classical forms of spatial linguistic spread. The  wave  model describes the radiation of linguistic change outwards from a central focus. Changes can also jump between population centres in a process known as  hierarchical diffusion . It has recently been proposed that the spatial evolution of dialects can be understood using surface tension at linguistic boundaries. Here we show that the inclusion of long-range interactions in the surface tension model generates both wave-like spread, and hierarchical diffusion, and that it is surface tension that is the dominant effect in deciding the stable distribution of dialect patterns. We generalize the model to allow population mixing which can induce shrinkage of linguistic domains, or destroy dialect regions from within.

Unifying models of dialect spread and extinction using surface tension dynamics