Competitiveness Function for a Bilingual Community Model
Abstract
The purpose of this work is to construct competitiveness functions for the bilingual community model.
Materials and methods. The work uses a new model of a bilingual community, which takes into account: the effect of acquiring a second language at an early age; the effect of mutual assistance within a group of the same language. In the model, languages are characterized by parameters of prestige, the likelihood of language acquisition at an early age, the parameter of mutual assistance and the initial number of native speakers. The problem of determining the results of language competition based on their characteristic parameters is considered.
Results. A new method for solving the problem of the results of language competition is proposed. For this purpose, a new concept is introduced in linguistic dynamics: the competitiveness function. To restore the competitiveness function, a ranking method is used, which is related to dividing ordered pairs of languages (under fixed initial conditions) into two classes “the first language displaces the second” and “the second language displaces the first”. The competitiveness function is sought in the form of a power function depending on the language parameters. In this case, the values of the function coefficients are identified based on the processing of available data on the dynamics of the model. The values of the competitiveness functions are analyzed, the results are compared with the observed statistics, and on this basis a forecast is made for the further development of dynamics. The application of this technique is demonstrated on a model in which finding a solution in analytical form is difficult.
Conclusion. The proposed methodology for constructing the competitiveness function is quite general and can be applied to a wide range of models describing population dynamics. The forecast made on the basis of the constructed competitiveness functions agrees well with empirical data.
References
X. Castellу, V. Eguiluz, and M. San Miguel, “Ordering Dynamics with Two Non-Excluding Options: Bilingualism
in Language Competition,” New Journal of Physics, vol. 8, no. 12, p. 308, 2006; doi:10.1088/1367-
/8/12/308
J.Mira and Aˊ . Paredes, “Interlinguistic Similarity and Language Death Dynamics,” Europhysics Letters
(EPL), vol. 69, no. 6, pp. 1031–1034, 2005; doi:10.1209/epl/i2004-10438-4
I. Baggs and H. Freedman, “A mathematical model for the dynamics of interactions between a unilingual
and a bilingual population: persistence versus extinction,” J. Math. Sociol, vol. 16, no. 1, pp. 51–75,
; doi:10.1080/0022250X.1990.9990078
I. Baggs and H. Freedman, “Can the speakers of a dominated language survive as unilinguals?: A mathematical
model of bilingualism,” Math. Comput. Model., vol. 18, no. 6, pp. 9–18, 1993; doi:10.1016/0895-
(93)90122-F
J. Wyburn and J.Hayward, “The future of bilingualism: an application of the Baggs and Freedman
model,” J. Math. Sociol, vol. 32, no. 4, pp. 267–284, 2008; doi:10.1080/00222500802352634
M. Diˊ az and J. Switkes, “A mathematical model of language preservation,” vol. 7(5), p. e069752021,
; doi:10.1016/j.heliyon.2021.e06975
D. Abrams and S. Strogatz, “Modelling the Dynamics of Language Death,” Nature, vol. 424, no. 6951,
pp. 900; doi:10.1038/424900a
J. Birch, “Natural selection and the maximization of fitness,” Biol. Rev., vol. 91, no. 3, pp. 712–727, 2016.
O. Kuzenkov, A. Morozov, and G. Kuzenkova, “Recognition of patterns of optimal diel vertical migration
of zoo-plankton using neural networks,” in Proc. of IJCNN 2019 – Int. Joint Conf. on Neural Networks.
Budapest. Hungary. July 14–19, 2019, pp. 1–6, 2019.
O. Kuzenkov, Fitness as a general technique for modeling the processes of transmission of non-innate
information, N. Novgorod, Russia: UNN, 2019.
O. Kuzenkov, E. Ryabova, and K. Krupoderova, Mathematical models of selection processes, N. Novgorod,
Russia: Nizhny Novgorod State University, 2010 (in Russian).
A. Gorban, Equilibrium bypass, Novosibirsk, Russia: Science, 1984 (in Russian).
A. Gorban, “Selection Theorem for Systems with Inheritance,” Math. Model. Nat. Phenom., vol. 2, no. 4,
pp. 1–45, 2007 (in Russian).
A. Gorban, “Self-simplification in Darwin’s systems,” Lecture Notes in Computational Science and Engineering,
vol. 75, pp. 311–344, 2011.
G. Karev and I. Kareva, “Replicator equations and models of biological populations and communities,”
Math. Model. Nat. Phenom. vol. 9, pp. 68–95, 2014.
O. Kuzenkov and E. Ryabova, “Variational Principle for Self-replicating Systems,” Math. Model. Nat.
Phenom., vol. 10, no. 2, pp. 115–129, 2015.
O. Kuzenkov and E. Ryabova, “Limit possibilities of solution a hereditary control system,” Dif. Eq.,
vol. 51, no. 4, pp. 500–511, 2015.
O. Kuzenkov and A. Morozov, “Towards the Construction of a Mathematically Rigorous Framework
for the Modelling of Evolutionary Fitness,” Bull. Math. Biol., vol. 81, no. 11, pp. 4675–4700, 2019;
doi:10.1007/s11538-019-00602-3
M. Mohri, A. Rostamizadeh, and A. Talwalkar, Foundations of Machine Learning, Cambridge, MA, US:
The MIT press, 2018.
O. Kuzenkov, “Construction of the fitness function depending on a set of competing strategies based
on the analysis of population dynamics,” Izvestiya VUZ. Applied Nonlinear Dynamics, vol. 30, no. 3,
pp. 276–298, 2022 (in Russian); doi:10.18500/0869-6632-2022-30-3-276-298
O. Kuzenkov and E. Ryabova, “Optimal control of a hyperbolic system on a simplex,” Journal of computer
and systems sciences international, vol. 42, no. 2, pp. 227–233, 2003.
O. Kuzenkov, “The investigation of the population dynamics controls problems based on the generalized
Kolmogorov model,” Journal of computer and systems sciences international, vol. 48, no. 5, pp. 839–
, 2003. bibitemlib-23 O. Kuzenkov and E. Ryabova, “Optimal control for a system on a unit simplex
in infinite time,” Automation and remote control, vol. 66, no. 10, pp. 1594–1602, 2005.
N. Sh. Alexandrova, “Disappearance of languages and natural bilingualism,” Multilingualism and transcultural
practices, vol. 20, no. 3, pp. 436–455, 2023 (in Russian); doi:10.22363/2618-897x-2023-20-3-436-
N. Francis, MICHEL PARADIS, — A Neurolinguistic Theory of Bilingualism, Amsterdam, Netherlands:
J. Benjamins Publ., 2007; doi: 10.1017/s004740450721005x
N. Alexandrova, “Native language, foreign language and linguistic phenomena that have no name,”
Questions of linguistics, vol. 3, pp. 88–100, 2004 (in Russian).
I. Ba, P. I. Ndiaye, M. Ndao, and A. Diakhaby, “An Extension of Two Species Lotka-Volterra Competition
Model,” Biomath Communications, vol. 8, no. 2, p. 2112171, 2022; doi: 10.11145/bmc.2021.12.171
A. Medvedev and O. Kuzenkov, “Generalization of the Abrams–Strogatti model of language dynamics
to the case of several languages,”in Proc. XXI International Scientific Conference on Differential,
Mogilev, May 23–27, 2023, vol. 2, pp. 117–118, 2023.
C. Sutantawibul, P. Xiao, S. Richie, and D. Fuentes-Rivero, “Revisit Language Modeling Competition
and Extinction: A Data-Driven Validation,” Journal of Applied Mathematics and Physics, vol. 06, no. 07,
pp. 1558–1570, 2018; doi: 10.4236/jamp.2018.67132
M. Zhang and T. Gong, “Principles of parametric estimation in modeling language competition,” Proc.
of the NationalAcademy of Sciences, vol. 110, no. 24, pp. 9698–9703, 2013; doi: 10.1073/pnas.1303108110
J. A. van Leuvensteijn, M. C. Tooren, W. J. J. Pijnenburg, and J. M. van der Horst, eds., “Geschiedenis
van de Nederlandse taal,” Amsterdam, Netherlands: Language Faculty of Humanities. Amsterdam
University Press, 1997 (In Dutch).
This work is licensed under a Creative Commons Attribution 4.0 International License.
