Threshold model of language competition including the bilingual state

TL;DR

This work extends threshold-based concepts from two-state language models to a three-state framework that explicitly includes bilingual speakers. By introducing memory/learning thresholds into both language learning and attrition, and allowing bilinguals to influence effective speaker fractions, the authors derive a piecewise-linear dynamical system that is analytically tractable. They classify a rich set of long-term behaviors into pure, intermediate, and limiting regimes, including extinction, frozen neutrality, and stable coexistence with diverse mono- and bilingual compositions; in symmetric cases, 14 topologies emerge with five principal attractors. The results reveal that thresholding can produce robust coexistence and long-lived intermediate states, offering qualitative alignment with observed bilingual communities and providing a basis for exploring policy and structural changes in multilingual societies. The framework is general and potentially applicable to other exposure-dependent acquisition and retention processes beyond language dynamics.

Abstract

We propose a threshold model of language competition which includes intermediate bilingual state. The model is based on the Minett-Wang model but through the introduction of thresholds in the language shift rates it incorporates the effects of memory and learning. The model is piecewise-linear, allowing the exact analytical treatment. We study the symmetric case where two competing languages are equivalent in terms of status and social pressure and provide a complete list of the various dynamical regimes. We also study several limiting regimes corresponding to asymmetric systems and characterize the full spectrum of possible asymptotic behaviors. Unlike the Minett-Wang model, which always predicts the extinction of one of the languages, the proposed new model exhibits a wide range of possible equilibrium scenarios, including equilibrium states of coexistence. Most commonly, in such coexistence regimes the minority language speakers are either completely monolingual or completely bilingual.

Figures (10)

• Figure 1: Panels (A) and (B) represent the state of a three-state system in Cartesian coordinates and by a ternary diagram, respectively. Panel (C): learning and attrition thresholds \ref{['Lrates-fractions-1']} and \ref{['eq_newrates-attrit']} correspond to straight lines, each of them splits the ternary diagram into regions where the corresponding process is on or off; Panel (D): whenever parameters $\beta^+$ and $\beta^-$ are unequal, the learning and retention thresholds may intersect; if $\beta^- > \beta^+$ the steeper line corresponds to the retention threshold.
• Figure 2: Panel (A): normalized learning rate, $r_{ Y \to Z}/k_{YZ}$, versus the corresponding effective fraction of X-speakers, $x_\mathrm{eff}^+$, needed to learn language X. Panel (B): normalized attrition rate, $r_{ Z\to Y}/k_{ZY}$, versus the corresponding effective fraction of X-speakers, $x_\mathrm{eff}^-$, needed to maintain language X. Different curves correspond to different three-state models: dashed black curves---MW bilinguals model with volatility $a>1$, Eqs. \ref{['Lrates-2']}; thin blue solid curves---threshold model with step function rates, Eqs. \ref{['rates']}; thick red solid curves---heterogeneous threshold model with sigmoidal shape. A similar scheme holds for the Y-language.
• Figure 3: Dynamics of system \ref{['system']} in the case when each reaction channel is either permanently open (red arrow) or permanently closed (gray arrow). The bold dots and lines on the ternary plots indicate the attractors. Thin lines correspond to a schematic representation of evolution trajectories and arrows on them indicate the direction of the evolution. Gray area is the area of neutral stability. The exemplary trajectories are drawn for a fully symmetric case $\gamma_x = \gamma_y$, $\kappa = 1/2$. The last column introduces color-coding of the states, which is used throughout the following figures. All states except 1, 3, 7 and 10 have dual counterparts, which can be obtained by $X \longleftrightarrow Y$ replacement and are not shown to save the space.
• Figure 4: Example of dynamics with $x^*_+ =y^*_+= 0.3, x^*_- =y^*_- = 0.15, \kappa = 1/2$. (A) The switching conditions split the whole space of possible compositions into 7 areas -- numbers and colors correspond to the pure states introduced in Fig. \ref{['fig:pure']} and in the text; (B) schematic depiction of the dynamic flows in the system -- red dashed lines separate basins of attraction; (C) attractors (black bold lines and points) of Eqs. (\ref{['system']}) for this particular set of parameters and boundaries of their basins of attraction; red dots and lines correspond to the initial conditions and trajectories of the dynamics shown in panels (D)-(F); (D)-(F) examples of dynamics for three different sets of initial conditions for $\gamma_{x,y}=0.1$; red, green and orange lines show the time evolutions of $x(t),y(t),z(t)$ given by \ref{['system']} with Heaviside (full lines) and S-shaped (\ref{['sshape']}, dashed lines) rates; the dashed vertical lines indicate the switching between the regimes, with the numbers inside the panels indicating the regimes; (D) initial condition $(x_0,y_0,z_0)=(0.75,0.25,0)$ converging to the extinction of language Y, $(x_\infty,y_\infty,z_\infty)=(1,0,0)$; (E) evolution starting from initial condition $(x_0,y_0,z_0)=(0.6,0.4,0)$ converging to a long-term coexistence of monolinguals of X and bilinguals , $(x_\infty,y_\infty,z_\infty)=(0.45,0.55,0)$;, (F) evolution starting from initial condition $(x_0,y_0,z_0)=(0.1,0.15,0.75)$ is frozen from the start, so $(x_\infty,y_\infty,z_\infty)=(0.1,0.15,0.75)$.
• Figure 5: Flows and attractors of the system with no language attrition. Top row: $x^*_+ + y^*_+>1$; bottom row $x^*_+ + y^*_+ \leq1$. First column shows splitting of the space of parameters into regions where the conditions in Eqs. \ref{['system_noattr']} are opened differently; second column is a flaw diagram; third column shows the attractors of the dynamics. Note that the shape of the line splitting the basins of attraction in the bottom raw is, generally speaking, $\kappa$-dependent and it is a straight line going through $z=1$ angle of the ternary diagram only if $\kappa=1/2$. The gray point inside the triangle in the bottom row corresponds to a partially stable stationary point at $(x,y,z)=(x_+^*,y_+^*,1-x^*_+-y_+^*)$, it is attractive along the vertical black line, repulsive in perpendicular direction, neutral if approached from the gray area side.