1. Glossary of Technical Terms
Volatility: The willingness of individuals in a society to learn a
language.
Prestige: The value of a language to a society as a whole.
Project Description
• Determine a model that stabilizes the coexistence of languages with
meaningful applications
• Understand the effect of language social standing and resistance to
assimilation on the current integration of language communities within a
bilingual model of language competition
• Understand the effect of temporal and spatial parameters on the social
standing and resistance to assimilation of a language community within
a bilingual model of language competition
Scientific Challenges
• Current literature reflects the Abrams-Strogatz model and its variants,
where one language always dominates
Potential Applications
• A method for prolonging the existence of a minority language
• A method for effectively modelling oscillating temporal and spatial
parameters within a given model
Modeling Language Competition: Coexistence of Languages
Team Members: Sana Shahid ● Rema Hamdan ● Catherine Gallardo ● Kevin Lendo
Original Numerical dynamics show language death
Methodology
1. Begin with the Bilinguals Minett-Wang model, as an extension of the
Abrams-Strogatz Model:
𝑝𝑖, 𝐴 → 𝐴𝐵 = 1 − 𝑠 𝜎𝑖, 𝐵
𝑎
, 𝑝𝑖, 𝐵 → 𝐴𝐵 = 𝑠 𝜎𝑖, 𝐴
𝑎
𝑝𝑖, 𝐴𝐵 → 𝐴 = 1 − 𝑠 1 − 𝜎𝑖, 𝐴
𝑎
, 𝑝𝑖, 𝐴𝐵 → 𝐵 = 𝑠 1 − 𝜎𝑖, 𝐵
𝑎
2. Alter prestige to be a function of space and time, with neutral volatility:
𝑠 𝑥, 𝑦, 𝑡 = 𝛽 sin 𝑐𝑥 sin 𝑑𝑦 + 𝑒−𝑡
sin(𝑡100𝑒 𝑡
) , 𝑎 = 1
3. Collect data
4. Alter volatility to be a function of space and time, with socially
equivalent languages:
𝑎 𝑥, 𝑦, 𝑡 = 𝛽 sin 𝑐𝑥 sin 𝑑𝑦 + 𝑒−𝑡sin(𝑡100𝑒 𝑡) , s =
1
2
5. Collect data
6. Alter prestige to be a function of space ad volatility to be a function of
time:
𝑠 𝑥, 𝑦 = 𝛽 sin 𝑐𝑥 sin 𝑑𝑦 +
1
2
, 𝑎(𝑡) = 𝑒−𝑡sin(𝑡100𝑒 𝑡)
7. Collect data
Results
1. Start with The original model. Set volatility to 1 and prestige to ½.
2. The analysis of this model shows two stable fixed points indicating
language death, as well as an unstable fixed point.
1. The numeric results agree with the analytic findings, showing language
death.
1. Next experiment by setting values of prestige and volatility as functions
(5)
1. Change value of prestige to obtain seemingly stable
References
1. Castello, X. Equiluz, V.M Loureiro-Porto, l. San Miguel, The
Fate of Bilingualism in a Model of Language Competition
(2007)
2. Castello, X. Vasquiz, Agent Based Models of Language
Competitons: Macroscopic Descritpions and Order_Disoder
Transitions (2010)
3. PPLANE (http://math.rice.edu/~dfield/) is developed by
John C. Polking, Department of Mathematics, Rice
University.
Acknowledgments
This project was mentored by Colin Clark, whose help is
acknowledged with great appreciation.
Support from a University of Arizona TRIF (Technology
Research Initiative Fund) grant to J. Lega is also gratefully
acknowledged.
Dynamics of the Oregonator model [2], plotted with the software
PPLANE [3].
(1)
(2)
(3)
(4)
(5)
Second trial of Dynamics
β = 0.5, c = 0.5, d = 0.5
Third trial of Dynamics
β = 0.5, c = 0.1, d = 0.1
Initialization
1. To Consider physical dynamics, a two space (x,y) torus grid was used,
as the modeled society.
2. Additionally, time models exponential growth with units in terms of the
square of the elapsed time in years.
3. To main initial societies were used in order to eliminate the possibility
of initial location as a condition on the asymptotic behavior.