Microscopic Abrams–Strogatz model of language competition

Abstract

Abstract:The differential equations of Abrams and Strogatz for the competition between two languages are compared with agent-based Monte Carlo simulations for fully connected networks as well as for lattices in one, two and three dimensions, with up to 10 9 agents.Keywords: Monte Carlo, language competition Many computer studies of the competition between different languages, triggered by Abrams and Strogatz [1], have appeared mostly in physics journals using differential equations (mean field approximation [2, 3, 4, 5]) or agent-based simulations for many [6,7,8,9] or few [10, 11] languages. A longer review is given in [12], a shorter one in [13]. We check in this note to what extent the results of the mean field approximation are confirmed by agent-based simulations with many individuals. We do not talk here about the learning of languages [14,15].The Abrams-Strogatz differential equation for the competition of a language Y with higher social status 1 − s against another language X with lower social status s iswhere a ≃ 1.3 [1] and 0 < s ≤ 1/2. Here x is the fraction in the population speaking language X with lower social status s while the fraction 1−x speaks language Y. As initial condition we consider the situation in which both languages have the same number of speakers, x(t = 0) = 1/2. Figure 1 shows exponential decay for a = 1.31 as well as for the simpler linear case a = 1. For s = 1/2 the symmetric situation x = 1/2 is a stationary solution 1

Optimal language policy for the preservation of a minority language

“…Mira and Paredes (2005) introduce the concept of similarity between competing languages and prove that both languages can survive if they are close to each other. Stauffer et al (2007) propose microscopic or individual based versions of the AS model and apply simulation techniques instead of averaging over the whole population. Mira and Paredes (2005), Minett and Wang (2008), Heinsalu et al (2014) and others extend the A-S model by additionally considering bilinguals.…”

The Fate of Bilingualism in a Model of Language Competition

“…With the aim of elucidating possible mechanisms that could stabilize the coexistence of these languages, we wish to discuss the role of bilingual individuals and social structure in the process of language competition. In this way, this paper complements our previous analysis of the agent-based version of Abrams-Strogatz two-state model [12]. This paper is, then, structured as follows.…”

“…However, numerical results for the agent based model for the Abrams-Strogatz dynamics [12] indicate a very different behaviour due to finite size fluctuation effects, with one of the two languages eventually becoming dominant, as we discuss in the following sections. Generally speaking, mean field equations do not give an appropriate description of finite size communities, or societies which are not fully connected.…”

“…To summarize the effect of the bilingual agents, we must first recall the results obtained in the analysis of the agent based model for the AbramsStrogatz dynamics for equivalent languages, where bilinguals are not taken into account [12]. For a two-dimensional regular lattice, this study shows that the average interface density decays following a power law ρ ∼ t −0.11 (consitent with a logarithmic decay), the dependence of the lifetime with the system size scales as τ ∼ N ln(N ), and the domain boundaries follow noisy interface dynamics, without well defined boundaries of the monolingual domains.…”

“…This seminal work, as well as others along the same line [7,8], belongs to the general class of studies of population dynamics based on nonlinear ordinary differential equations for the populations of speakers of different languages. In addition, other studies implement discrete agent based models with speakers of many [9,10] or few languages [11,12], as reviewed in [9,13]. These studies consider social structures modelled by complex networks in which agents are connected with one another.…”