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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 𝒮 n -symmetric transcritical bifurcation as learning becomes more reliable.

Keywords

Steady State Dynamical System Dynamical Equation Language Acquisition Bifurcation Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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