Abstract
We extend the Abrams-Strogatz model for competition between two languages [Nature 424, 900 (2003)] to the case of n(>=2) competing states (i.e., languages). Although the Abrams-Strogatz model for n=2 can be interpreted as modeling either majority preference or minority aversion, the two mechanisms are distinct when n>=3. We find that the condition for the coexistence of different states is independent of n under the pure majority preference, whereas it depends on n under the pure minority aversion. We also show that the stable coexistence equilibrium and stable monopoly equilibria can be multistable under the minority aversion and not under the majority preference. Furthermore, we obtain the phase diagram of the model when the effects of the majority preference and minority aversion are mixed, under the condition that different states have the same attractiveness. We show that the multistability is a generic property of the model facilitated by large n.
Highlights
The consensus problem, in which we ask whether the unanimity of one among different competing states is reached, and its mechanisms are of interest in various disciplines including political science, sociology, and statistical physics
In models of consensus formation, it is usually assumed that each individual possesses one of the different states that can flip over time
We separately considered the effect of the majority preference (Sect. 3.1) and the minority aversion (Sect. 3.2)
Summary
The consensus problem, in which we ask whether the unanimity of one among different competing states (e.g., opinions) is reached, and its mechanisms are of interest in various. In models of consensus formation, it is usually assumed that each individual possesses one of the different states that can flip over time. The dynamics of the model is based on the majority preference, which is regarded as the minority aversion because there are just two competing languages. Several authors found that different languages can stably coexist in variants of the AS model. Two languages can coexist by spatial segregation in a model in which competition dynamics and spatial diffusion are combined [28, 29]. We extend the AS model to the case of competition among a general number of languages, denoted by n. Where β (≥ 0) and a − β (≥ 0) represent the strength of the majority preference and the minority aversion, respectively.
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