An analytic solution of a model of language competition with bilingualism and interlinguistic similarity

• We solve a mean field model that describes the fate of language competition.
• A set of three intuitive parameters determines the dynamics of the competing languages.
• The equilibrium points of the system are obtained and their stability is determined.
• A phase space is portrayed. For any two languages their evolution can be forecasted.
• Oscillating orbits are banned, this affects the way in which the languages evolve.

An in-depth analytic study of a model of language dynamics is presented: a model which tackles the problem of the coexistence of two languages within a closed community of speakers taking into account bilingualism and incorporating a parameter to measure the distance between languages. After previous numerical simulations, the model yielded that coexistence might lead to survival of both languages within monolingual speakers along with a bilingual community or to extinction of the weakest tongue depending on different parameters. In this paper, such study is closed with thorough analytical calculations to settle the results in a robust way and previous results are refined with some modifications. From the present analysis it is possible to almost completely assay the number and nature of the equilibrium points of the model, which depend on its parameters, as well as to build a phase space based on them. Also, we obtain conclusions on the way the languages evolve with time. Our rigorous considerations also suggest ways to further improve the model and facilitate the comparison of its consequences with those from other approaches or with real data.