Abstract <p>In this paper, we present a spatially extended identity-based model for language shift dynamics that balances sociolinguistic realism with mathematical tractability. This model improves upon the original framework by eliminating unrealistic assumption of monolingual groups of the non-dominant language and by incorporating spatial diffusion to accurately represent geographic diversity. We consider four subpopulations defined by language use and ethnic identity: monolinguals and bilinguals identifying with either the dominant or non-dominant language. The system is described by a set of reaction-diffusion equations on a two-dimensional domain subdivided into high-, mid-, and low-interaction regions to reflect varying degrees of language contact and mobility. Numerical simulations reveal complex spatiotemporal patterns, including the spread of dominant-language identifiers from high-rate interaction zones into mid-rate interaction regions. Notably, the model exhibits cyclic dynamics in population proportions, indicating potential for long-term coexistence rather than inevitable language extinction—a feature absent in classical models. These results highlight the importance of integrating identity negotiation and spatial structure into models of language dynamics. The proposed framework provides a rigorous basis for future studies seeking to explain mechanisms of language maintenance, coexistence, and shift in multilingual environments.</p>

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Numerical Simulation of a Spatially Extended Identity-Based Model for Language Shift Dynamics

  • U. S. Kalachikova,
  • V. I. Sivtseva,
  • A. V. Timofeeva

摘要

Abstract

In this paper, we present a spatially extended identity-based model for language shift dynamics that balances sociolinguistic realism with mathematical tractability. This model improves upon the original framework by eliminating unrealistic assumption of monolingual groups of the non-dominant language and by incorporating spatial diffusion to accurately represent geographic diversity. We consider four subpopulations defined by language use and ethnic identity: monolinguals and bilinguals identifying with either the dominant or non-dominant language. The system is described by a set of reaction-diffusion equations on a two-dimensional domain subdivided into high-, mid-, and low-interaction regions to reflect varying degrees of language contact and mobility. Numerical simulations reveal complex spatiotemporal patterns, including the spread of dominant-language identifiers from high-rate interaction zones into mid-rate interaction regions. Notably, the model exhibits cyclic dynamics in population proportions, indicating potential for long-term coexistence rather than inevitable language extinction—a feature absent in classical models. These results highlight the importance of integrating identity negotiation and spatial structure into models of language dynamics. The proposed framework provides a rigorous basis for future studies seeking to explain mechanisms of language maintenance, coexistence, and shift in multilingual environments.