Analysing Language Change Using ODEs, PDEs and SDEs

Matthew Roberts; James Burridge

doi:10.59973/emjsr.397

Authors

Matthew Roberts School of Mathematics and Physics, University of Portsmouth, PO1 3QL Portsmouth, UK
James Burridge School of Mathematics and Physics, University of Portsmouth, PO1 3QL Portsmouth, UK

DOI:

https://doi.org/10.59973/emjsr.397

Keywords:

Replicator dynamics, Reaction–diffusion equations, Stochastic differential equations, Language change

Abstract

This paper investigates the mathematical modelling of language change using ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). Beginning with replicator equations derived from evolutionary game theory, deterministic models are analysed using phase portraits, stability theory, Jacobian linearisation and simplex geometry to explain how competing linguistic variants evolve over time. The work is then extended to spatial reaction–diffusion models, where travelling waves, steady states and isogloss dynamics are used to examine the geographical spread of linguistic features. Finally, stochastic effects are incorporated through Wright–Fisher diffusion, allowing random fluctuations and mutation to be included within the modelling framework. Analytical methods are complemented by
numerical simulations to illustrate the qualitative behaviour of each model. Together, these deterministic, spatial and stochastic approaches provide a mathematical framework for understanding how linguistic variation emerges, propagates and stabilises across populations and geographical regions.

References

[1] William Labov, Sharon Ash, and Charles Boberg. The Atlas of North American English. Mouton de Gruyter, 2006. DOI: https://doi.org/10.1515/9783110167467

[2] Peter Trudgill. Dialects in Contact. Blackwell, 1986.

[3] Marco Patriarca and Teemu Lepp¨anen. “Modeling language competition.” Physica A, 338(1–2):296–299, 2004. DOI: https://doi.org/10.1016/j.physa.2004.02.056

[4] Howard Giles, Nikolas Coupland, and Justine Coupland. “Accommodation Theory.” In Contexts of Accommodation, pages 1–68. Cambridge University Press, 1991. DOI: https://doi.org/10.1017/CBO9780511663673.001

[5] Warren J. Ewens. Mathematical Population Genetics. Springer, 2nd edition, 2004. DOI: https://doi.org/10.1007/978-0-387-21822-9

[6] Gareth J. Baxter, Richard A. Blythe, William Croft, and Alan J. McKane. “Utterance selection model of language change.” Physical Review E, 73(4):046118, 2006. DOI: https://doi.org/10.1103/PhysRevE.73.046118

[7] Peter D. Taylor and Leo B. Jonker. “Evolutionary stable strategies and game dynamics.” Mathematical Biosciences, 40(1–2):145–156, 1978. DOI: https://doi.org/10.1016/0025-5564(78)90077-9

[8] Josef Hofbauer and Karl Sigmund. Evolutionary Games and Population Dynamics. Cambridge University Press, 1998. DOI: https://doi.org/10.1017/CBO9781139173179

[9] Lawrence Perko. Differential Equations and Dynamical Systems. Springer, 3rd edition, 2001. DOI: https://doi.org/10.1007/978-1-4613-0003-8

[10] James D. Murray. Mathematical Biology II: Spatial Models and Biomedical Applications. Springer, 3rd edition, 2003. DOI: https://doi.org/10.1007/b98869

[11] Nigel Goldenfeld. Lectures on Phase Transitions and the Renormalization Group. Addison-Wesley, 1992.

[12] James A. Sethian. Level Set Methods and Fast Marching Methods. Cambridge University Press, 1999. DOI: https://doi.org/10.1137/S0036144598347059

[13] Bernt Øksendal. Stochastic Differential Equations: An Introduction with Applications. Springer, 6th edition, 2003. DOI: https://doi.org/10.1007/978-3-642-14394-6

[14] Drew Fudenberg and Christopher Harris. “Evolutionary dynamics with stochastic shocks.” Journal of Economic Theory, 57(2):420–441, 1992. doi:10.1016/0022-0531(92)90041-D. DOI: https://doi.org/10.1016/0022-0531(92)90044-I

[15] Hannes Risken. The Fokker–Planck Equation: Methods of Solution and Applications. Springer, Berlin, 2nd edition, 1996. DOI: https://doi.org/10.1007/978-3-642-61544-3

[16] Mark I. Freidlin and Alexander D. Wentzell. Random Perturbations of Dynamical Systems. Springer, New York, 3rd edition, 2012. DOI: https://doi.org/10.1007/978-3-642-25847-3

[17] Peter E. Kloeden and Eckhard Platen. Numerical Solution of Stochastic Differential Equations. Springer, Berlin, 1992. DOI: https://doi.org/10.1007/978-3-662-12616-5