Journal of Mathematical Biology

, Volume 46, Issue 3, pp 265–285 | Cite as

Bifurcation analysis of the fully symmetric language dynamical equation

  • W. Garrett Mitchener

Abstract

 In this paper, I study a continuous dynamical system that describes language acquisition and communication in a group of individuals. Children inherit from their parents a mechanism to learn their language. This mechanism is constrained by a universal grammar which specifies a restricted set of candidate languages. Language acquisition is not error-free. Children may or may not succeed in acquiring exactly the language of their parents. Individuals talk to each other, and successful communication contributes to biological (or cultural) fitness. I provide a full bifurcation analysis of the case where the parameters are chosen to yield a highly symmetric dynamical system. Populations approach either an incoherent steady state, where many different candidate languages are represented in the population, or a coherent steady state, where the majority of the population speaks a single language. The main result of the paper is a description of how learning reliability affects the stability of these two kinds of equilibria. I rigorously find all fixed points, determine their stabilities, and prove that all populations tend to some fixed point. I also demonstrate that the fixed point representing an incoherent steady state becomes unstable in an 𝒮n-symmetric transcritical bifurcation as learning becomes more reliable.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • W. Garrett Mitchener
    • 1
  1. 1.Program in Applied and Computational Mathematics, Fine Hall, Washington Road, Princeton, NJ 08544-1000. e-mail: wmitchen@princeton.eduUS

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