The Persistence of a Local Dialect When a National Standard Language is Present: An Evolutionary Dynamics Model of Cultural Diversity

Abstract

In recent decades, cultural diversity loss has been a growing issue, which can be analyzed mathematically through the use of the formalism of the theory of cultural evolution. We here study the evolutionary dynamics of dialects in order to find the key processes for mitigating the loss of language diversity. We define dialects as different speech systems of the same language which are mutually intelligible. Specifically, we focus on the survival of a local dialect when competing against a national standard language, with the latter giving an advantage in occupational and economic contexts. We assume individuals may use different dialects, in response to two different situations: they may use the national language in a formal workplace, while they may use a local dialect in family or close friend meetings. We consider the choice of a dialect is guided by two forces: (1) differential attractiveness of the local/standard language and (2) willingness to speak the same dialect (conformity factor) inside a private group. We found that the evolutionary outcome critically depends on how conformity works. Conformity enhances the effect of differential attractiveness between the local dialect and the standard language if conformity works favoring only those states in which all speakers use the same dialect (unanimity pressure model), but conformity has no effect at all if it works in proportion to the fraction among peers (peer pressure model).

A Private Language Dynamics: Peer Pressure Model

Here we discuss the general case for any group size $n > 2$.

As mentioned in the main text, the contribution of the conformity effect for an A speaker (to conform to the dialect B) inside a group with m speakers of the A dialect is given by $\xi (n-m)/(n-1)$. On the other hand, a B speaker inside the same group suffers the conformity pressure $\xi m/(n-1)$ to start speaking the standard language A.

With this in mind, we can generalize the differential equations for any $n \ge 2$, as shown below:

$$\begin{aligned} \frac{{\hbox {d}}z_0}{{\hbox {d}}t}&= -n \pi _\mathrm{a} z_0 + \left( \pi _\mathrm{b} + \xi \right) z_1 \end{aligned}$$

(15a)

$$\begin{aligned} \frac{{\hbox {d}}z_j}{{\hbox {d}}t}&= -\left[ \left( n-j\right) \pi _\mathrm{a} + j\pi _\mathrm{b} + \frac{2j\left( n-j\right) \xi }{n-1}\right] z_j \nonumber \\&\quad + \left( n-j+1\right) \left( \pi _\mathrm{a} + \frac{\left( j-1\right) \xi }{n-1}\right) z_{j-1} \nonumber \\&\quad + \left( j+1\right) \left( \pi _\mathrm{b} + \frac{\left( n-j-1\right) \xi }{n-1}\right) z_{j+1} \end{aligned}$$

(15b)

$$\begin{aligned} \frac{{\hbox {d}}z_n}{{\hbox {d}}t}&= -n \pi _\mathrm{b} z_n + \left( \pi _\mathrm{a} + \xi \right) z_{n-1} \end{aligned}$$

(15c)

with $j=1,2,\ldots ,n-1$.

Thus, we can calculate ${\hbox {d}}z_\mathrm{A}/{\hbox {d}}t$ by using the definition $z_\mathrm{A} = \dfrac{1}{n}\displaystyle \sum \nolimits _{j=1}^n j z_j$:

$$\begin{aligned} \begin{aligned} \frac{{\hbox {d}}z_\mathrm{A}}{{\hbox {d}}t} = \frac{{\hbox {d}}}{{\hbox {d}}t}\left( \frac{1}{n}\sum \limits _{j=1}^n j z_j\right) = \frac{1}{n}\sum \limits _{j=1}^n j \frac{{\hbox {d}}z_j}{{\hbox {d}}t} \end{aligned} \end{aligned}$$

(16)

Replacing the terms from Eqs. (15b) and (15c), we get the following equation:

$$\begin{aligned} \begin{aligned} \frac{{\hbox {d}}z_\mathrm{A}}{{\hbox {d}}t} =&\,\frac{-1}{n}\sum \limits _{j=1}^{n-1} j\left[ \left( n-j\right) \pi _\mathrm{a} + j\pi _\mathrm{b} + \frac{2\left( n-j\right) j\xi }{n-1} \right] z_j \\&+ \frac{1}{n}\sum \limits _{j=1}^{n-1} j\left( n-j+1\right) \left( \pi _\mathrm{a}+ \frac{\left( j-1\right) \xi }{n-1} \right) z_{j-1} \\&+ \frac{1}{n}\sum \limits _{j=1}^{n-1} j\left( j+1\right) \left( \pi _\mathrm{b}+\frac{\left( n-j-1\right) \xi }{n-1} \right) z_{j+1} \\&- n\pi _\mathrm{b} z_n + \left( \pi _\mathrm{a}+\xi \right) z_{n-1} \end{aligned} \end{aligned}$$

(17)

We can make a simple variable change in the last two summation terms so that they are also written in function of $z_j$. This change yields:

$$\begin{aligned} \begin{aligned} \frac{{\hbox {d}}z_\mathrm{A}}{{\hbox {d}}t} =&\,\frac{-1}{n}\sum \limits _{j=1}^{n-1} j \left[ \left( n-j\right) \pi _\mathrm{a} + j\pi _\mathrm{b} + \frac{2\left( n-j\right) j\xi }{n-1} \right] z_j \\&+ \frac{1}{n}\sum \limits _{j=0}^{n-2} \left( j+1 \right) \left( n-j\right) \left( \pi _\mathrm{a}+ \frac{j \xi }{n-1} \right) z_j \\&+ \frac{1}{n}\sum \limits _{j=2}^n \left( j-1\right) j \left( \pi _\mathrm{b}+\frac{\left( n-j\right) \xi }{n-1} \right) z_j \\&- n\pi _\mathrm{b} z_n + \left( \pi _\mathrm{a}+\xi \right) z_{n-1} \end{aligned} \end{aligned}$$

(18)

Thus, we have some expression which can be written as:

$$\begin{aligned} \frac{{\hbox {d}}z_\mathrm{A}}{{\hbox {d}}t} = \sum \limits _{j=2}^{n-2} a_j z_j + C \end{aligned}$$

(19)

The coefficients $a_j$ for $z_j$, for $j=2,3,\ldots ,n-2$ can be calculated from Eq. (18):

$$\begin{aligned} \begin{aligned} n a_j =&-j\left( n-j \right) \pi _\mathrm{a} - j^2 \pi _\mathrm{b} - \frac{2\left( n-j\right) j^2\xi }{n-1} + \left( n-j\right) \left( j+1\right) \pi _\mathrm{a} \\&+ \frac{\left( n-j\right) j \left( j+1 \right) \xi }{n-1} + \left( j-1\right) j \pi _\mathrm{b} + \frac{\left( n-j \right) j \left( j-1\right) \xi }{n-1} \end{aligned} \end{aligned}$$

(20)

Hence, after some algebraic manipulations, we can find that the expression for $a_j$ is given by:

$$\begin{aligned} a_j = \frac{1}{n} \left[ \left( n-j \right) \pi _\mathrm{a} - j\pi _\mathrm{b}\right] \end{aligned}$$

(21)

The independent term C is formed from the terms outside the summation term (for $j=2,3,\ldots ,n-2$), cf. Eq. (19), and can be written as:

$$\begin{aligned} \begin{aligned} nC =&-\left[ \left( n-1 \right) \pi _\mathrm{a} + \pi _\mathrm{b} + 2\xi \right] z_1 - \left( n-1\right) \left[ \pi _\mathrm{a} + \left( n-1 \right) \pi _\mathrm{b} + 2\xi \right] z_{n-1} \\&+ n \pi _\mathrm{a} z_0 + 2 \left[ \left( n-1 \right) \pi _\mathrm{a} + \xi \right] z_1 + \left( n-2\right) \left( n-1\right) \left( \pi _\mathrm{b} + \frac{\xi }{n-1} \right) z_{n-1} \\&+ \left( n-1\right) n \pi _\mathrm{b} z_n - n^2 \pi _\mathrm{b} z_n + n \left( \pi _\mathrm{a} + \xi \right) z_{n-1} \end{aligned} \end{aligned}$$

(22)

Since $z_0 = 1 - \sum \nolimits _{j=1}^n z_j$, we can group the terms for each $z_i$, getting the following expression for C:

$$\begin{aligned} \begin{aligned} C&= \pi _\mathrm{a} - \left( \pi _\mathrm{a} +\pi _\mathrm{b} \right) \left[ z_1 + \left( n-1 \right) z_{n-1} + n z_n\right] -\pi _\mathrm{a}\sum \limits _{j=2}^{n-2}z_j \end{aligned} \end{aligned}$$

(23)

By replacing Eqs. (21) and (23) into Eq. (19), we get:

$$\begin{aligned} \frac{{\hbox {d}}z_\mathrm{A}}{{\hbox {d}}t} = - \sum \limits _{j=1}^n \frac{j\left( \pi _\mathrm{a} + \pi _\mathrm{b}\right) z_j}{n} + \pi _\mathrm{a} \end{aligned}$$

(24)

Finally, using the definition for $z_\mathrm{A}$, the final expression for ${\hbox {d}}z_\mathrm{A}/{\hbox {d}}t$ can be written as:

$$\begin{aligned} \frac{{\hbox {d}}z_\mathrm{A}}{{\hbox {d}}t} = -\left( \pi _\mathrm{a} + \pi _\mathrm{b} \right) z_\mathrm{A} + \pi _\mathrm{a} \end{aligned}$$

(25)

From Eq. (25), we can conclude that for any $n>2$, there is no dependence on $\xi$ in both the equilibrium point and the dynamics of $z_\mathrm{A}$, the average fraction of A speakers in a population divided into groups of n individuals.