Utterance selection model of language change

We present a mathematical formulation of a theory of language change. The theory is evolutionary in nature and has close analogies with theories of population genetics. The mathematical structure we construct similarly has correspondences with the Fisher-Wright model of population genetics, but there are significant differences. The continuous time formulation of the model is expressed in terms of a Fokker-Planck equation. This equation is exactly soluble in the case of a single speaker and can be investigated analytically in the case of multiple speakers who communicate equally with all other speakers and give their utterances equal weight. Whilst the stationary properties of this system have much in common with the single-speaker case, time-dependent properties are richer. In the particular case where linguistic forms can become extinct, we find that the presence of many speakers causes a two-stage relaxation, the first being a common marginal distribution that persists for a long time as a consequence of ultimate extinction being due to rare fluctuations.

Figure 1

(Color online) Speakers in the society interact with different frequencies (shown here schematically by different thicknesses of lines connecting them). The pair of speakers $i,j$ is chosen to interact with probability $G_{ij}$.

Figure 2

(Color online) Both speakers $i$ and $j$ produce an utterance, with particular lingueme variants appearing with a frequency given by the value stored in the utterer’s grammar when no production biases are in operation. In this particular case three variants are shown ($\alpha ,\beta$ and $\gamma$) and the number of tokens, $T$, is equal to 6.

Figure 3

(Color online) In the biased reproduction model, the probability of uttering a particular variant is a linear combination $M$ of the values stored in the grammar.

Figure 4

(Color online) After the utterances have been produced, both speakers modify their grammars by adding to them the frequencies with which the variants were reproduced in the conversation. Note each speaker retains both her own utterances as well as those of her interlocutor, albeit with different weights.

Figure 5

(Color online) A generation of a population of utterances in the utterance selection model could be defined as the set of tokens produced by all speakers in the macroscopic time interval $\mathrm{\Delta }t$.

Figure 6

(Color online) Fisher-Wright “beanbag” population genetics. The population in generation $t+1$ is constructed from generation $t$ by (i) selecting a gene from the current generation at random; (ii) copying this gene; (iii) placing the copy in the next generation; (iv) returning the original to the parent population. These steps are repeated until generation $t+1$ has the same-sized population as generation $t$.

Figure 7

Time development of the exact solution $P(x,t)$ of the Fokker-Planck equation for a single speaker with two variants initially, when bias is absent and $x_0=0.7$.

Figure 8

The stationary distribution with one speaker and two variants for $m_1=m_2=0.2$.

Figure 9

The stationary distribution with one speaker and two variants for $m_1=0.8$ and $m_2=0.6$.

Figure 10

The time development of the mean of a single speaker $⟨x_i⟩$ for two different choices of mutation parameters. In each case $x_{i,0}=0.7$, $N=10$, and $h=0.5$. $T=1$. The overall population mean $⟨x⟩$ is shown as a dashed line for comparison, with $x_0=0.3$.

Figure 11

The single-speaker marginal stationary distribution when $N=10$, $h=0.2$, and $m_1=m_2=0.2$. Bars are the distribution obtained from simulation, while the curve is the approximate $\beta$ distribution.

Figure 12

The mutation value $m_c$ at which the stationary probability distribution function of $x_i$ changes from a concave to a convex distribution, as a function of $h$ for $N=2$, 10, and 100. Mutation is assumed symmetric: $m_1=m_2$.

Figure 13

The average speaker stationary distribution when $N=10$, $h=0.2$, and $m_1=m_2=0.2$. Bars are the distribution obtained from simulation, while the curve is the approximate $\beta$ distribution.

Figure 14

The mean time to fixation to each boundary as a function of $h$, for a system with 20 speakers and $x_0=0.3$. The solid curves are for an inhomogeneous initial condition, and the dashed curves are for a homogeneous initial condition. The lower curves are ${\tau }_2$ and the upper curves are ${\tau }_1$.

Figure 15

The distribution of speaker grammar values over a time series, for (top) an ensemble of realizations (none of which reach fixation during the period shown) and (bottom) the analytic $\beta$ distribution approximation, both for $N=20$ and $h=0.2$.

Figure 16

The number of realizations remaining unfixed at time $t$, with initially 1000 realizations. Dashed curve is $1000e^{-t∕\tau }$ where $\tau$ is given by Eq. (70).

Figure 17

The distribution of speaker grammar values over a time series, for (top) an ensemble of realizations (including fixing realizations) and (bottom) the analytic $\beta$ distribution approximation, both for $N=20$ and $h=0.2$.