Learning thresholds lead to stable language coexistence

Abstract

We introduce a language competition model that is based on the Abrams-Strogatz model and incorporates the effects of memory and learning in the language shift dynamics. On a coarse grained time scale, the effects of memory and learning can be expressed as thresholds on the speakers fractions of the competing languages. In its simplest form, the resulting model is exactly solvable. Besides the consensus on one of the two languages, the model describes additional equilibrium states that are not present in the Abrams-Strogatz model: a stable dynamical coexistence of the two languages and a frozen state coinciding with the initial state. We show numerically that these results are preserved for threshold functions of a more general shape. The comparison of the model predictions with historical datasets demonstrates that while the Abrams-Strogatz model fails to describe some relevant language competition situations, the proposed model provides a good fitting.

Figures (6)

• Figure 1: Panel (a): Transition rates $r_x(x)$. Black dotted line -- power law rate \ref{['rates']} of the AS model with $a = 1.3$; thin blue line -- step function rate defined by Eqs. \ref{['step']}-\ref{['const']}; thick red line -- generalized sigmoid rate (\ref{['SShape_replacement']}) with $w = 0.1$ and constant $r_{0x}$; dashed magenta line -- generalized sigmoid rate with $w = 0.1$ and $r_{0x}(x) = 0.25 + 0.75x$. For all transition rates $j_x = 1, x^* = 0.2$. Panel (b): The $x$-velocity field defined by Eq. \ref{['AS0']}, computed for the same transition rates as in panel (a). We have assumed $y^* = x^*, \gamma = 0.55$.
• Figure 2: Semi-logarithmic plot of the fraction of speakers of Welsh in Wales [panel (a)] and Swedish in Finland [panel (b)]. The points corresponding to census data walesfinland are well fitted by the solution \ref{['coexponent']} of the proposed model (thick black lines). Thin red lines correspond to the best fits of the AS model with $a \geq 1$. See Appendix C3 for the details of the fits .
• Figure 3: Panel (a): Plane $x^*$-$y^*$ split into different regions corresponding to possible outcomes of the dynamics defined by Eq. \ref{['gAS1']}: state 0 represents a dynamics frozen at the initial condition $x_0 = x(0)$; state I corresponds to the X-consensus attractive fixed point $x_\text{I} = 1$; state II -- to Y-consensus point $x_\text{II} = 0$; state III -- to the coexistence point $x_\text{III} = \gamma$. Panel (b): $x_0$ vs $x^*(y^*)$ cross-section of the $x_0$-$x^*$-$y^*$ space along the dashed line $x^* = y^*$ in panel (a), showing the dependence on the initial conditions $x = x(0)$: each region leads to the unique asymptotic solution indicated, apart from the shaded region, defined by $x^* = y^* = \min(\gamma, 1 - \gamma)$, $x^* < x_0 < 1 - x^*$, leading to $x_\text{II} = 0$ if $\gamma < 1/2$ or $x_\text{I} = 1$ if $\gamma > 1/2$.
• Figure 4: Time-evolution of the fraction of X-speakers for different initial conditions $x(0) = 1/8, 2/8, \dots, 7/8$ (from red to purple) and thresholds $x^* = y^*$; $\gamma = 0.3$. Solid lines correspond to the step-function rates, dashed lines to the sigmoid rates \ref{['SShape_replacement']} with $w=0.07$. Panel (a): $x^* = 0.7$; possible outcomes are I, II and 0 [see Fig. \ref{['Diagram']}(b)]; for sigmoid rates the 0-regime becomes unstable and $x$ converges to either $x_{\text{I}}$ or $x_{\text{II}}$. Panel (b): $x^* = 0.35$, possible outcomes are I, II; for step-function rates, there is a discontinuity in $\dot{x}$ when crossing the value $x = x^*$. Panel (c): $x^* = 0.2$, possible outcomes are I, II and III. Using sigmoid rates, the boundaries of the basins of attraction move: e.g., in panel (c) the initial condition $x_0 = 7/8$ is attracted to the intermediate stable point $x_{\text{III}}$; however, outcome I is still possible, as illustrated by trajectory with $x_0 = 15/16$ (magenta dashed line).
• Figure 5: Data on the time evolution of the share of minority-language speakers in various countries: Welsh in Wales (blue circles), Swedish in Finland (orange squares), Quechua in Peru (green diamonds), English in Quebec (red triangles), French in the whole of Canada (purple downward triangles), Russian in Estonia (brown empty circles). Connecting dashed lines are guides to the eye.