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its mirrored variants !′ and !′′, were unstable in both high and low volatility states. They
however are crucial to understanding the model’s dynamics, as is shown in the following
Tallinna Ülikooli üliõpilaste 2015/2016. õppeaasta PARIMAD TEADUSTÖÖD / Artiklite kogumik HARIDUSTEADUSED section.
1.3 Dynamics of the three-language model
High volatility can be interpreted as a society where individuals change their chosen language
very frequently. The stable solution in this case is and the coordinates , and , given !!!!
in (8), determine the fraction of speakers each language will hold after the model has relaxed. Since they don’t depend on initial values of speakers, the success of a language is largely determined by the relation of its prestige to volatility.
We found the effect of prestige and volatility on the dynamics of the model to be summarisable as follows: In high volatility states (a < 1), all linguistic groups survive and the number of agents they hold after the model’s relaxation is determined by volatility and prestige. As volatility nears 0, prestige has a strong influence on the number of agents held. With volatility increases toward 1, prestige loses influence.
For low volatility states, the stable points are , and , but just one language survives.
Which one of the three points is chosen, is determined by prestige and volatility, through the stationary points , ′, ′′ and . In these zones, only one of the first three stationary
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points is stable – one at the corner of the phase. This is visible in Fig. (1), where we plot x(t) vs y(t), thus all coordinates represent a distribution of speakers in time. All trajectories in Fig. (1) start from the center and move toward the edges.
Some relaxation trajectories on Fig. (1) exhibit non-monotonic relaxation. Trajectories that move close to the phase lines, in some cases have two of the three participants gaining speak- ers in time. This means that during the process itself, a language might appear “healthy” and growing, but in reality, as soon as the weaker competitor is gone, it will be consumed by the
dominant language as well. Some of these lines are noted in Fig. (1) by arrows.
cOncluSiOnS
Given that languages are going extinct rapidly in the modern era [1], and based on the multiple situations where a competition between three similar groups takes place, the three-language model is therefore a valuable tool in beginning their research. Further work with the model is necessary, especially in tting it to real data and testing its assumptions and predictions in prac- tise as well as the appearance of non-monotonic dynamics.
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