Another line of my work on social behaviors is in the emerging field of research on language dynamics or language evolution. A critical research subject in this field is on the dynamics of language death. For in recent years there have been many world languages disappearing due to rapid cultural mixes or changes. In 2003, Abrams and Strogatz proposed a mathematical model on the dynamics of language death and sparked years of research on this subject in mathematical and physical sciences.
My work on language dynamics follows the work by Abrams and Strogatz by reformulating the AS model as a population game. In this game, the speakers are assumed to decide how to use the languages for their best possible social benefits, which depend on the population size as well as the social status of the languages as described in the AS model: The smaller the population size, the less likely the language is selected, and the lower the social status, the more likely the language is abandoned. The languages of small population sizes or low social status are thus the most vulnerable in the struggling of language competition.
Indeed, based on the game model for language competition, it is easy to show that for a population of two languages, if the payoff for speaking one language is higher, the population size for speaking this language would increase; otherwise, it would decrease. When the game reaches an equilibrium, one language takes over the whole population while the other becomes extinct. There is an equilibrium state with none of the population sizes equal to zero, but it is unstable and hence unsustainable.
Nonetheless, the unstable state of the language game hints a population threshold for language growth: If the population size for one language passes a certain threshold, this language population will grow even if the social status of the language is low. If the population size is below the threshold, it will decrease even if the social status of the language is high.
In addition to population size and social status, many factors contribute to the growth or decline of languages such as language policy, education, immigration, family influence, etc. It can be proved that the game model for language competition can be modified to include these factors, and with these social influences, the payoffs for speaking the languages may vary in such a way that a stable equilibrium state can be reached with the languages co-existing in the population.