Threshold Model of Language Competition

Updated 14 November 2025

Threshold models of language competition are mathematical frameworks where discrete exposure and memory thresholds trigger abrupt shifts between language dominance, coexistence, or freezing.
They partition the state space into distinct regimes based on critical speaker fractions, yielding sharp bifurcations that accurately match historical language shift data.
These models provide practical insights into early-warning signals near tipping points and extend to bilingual and multi-language dynamics using both step and smooth sigmoid functions.

The threshold model of language competition encompasses a class of mathematical formulations in which the dynamics of language adoption, maintenance, or shift depend on surpassing critical fraction thresholds of exposure, memory, or social reinforcement. These thresholds generate sharp qualitative transitions, or bifurcations, in the stability and nature of multilingual equilibria, introducing regimes such as stable coexistence, freezing, or rapid extinction, not present in purely continuous-rate or power-law interaction models without threshold effects.

1. Model Foundations and Key Variables

In canonical threshold models, a well-mixed population of fixed size is partitioned according to linguistic affiliation, typically between two competing languages (X and Y). The principal dynamical variable is $x(t)$ , the fraction of X-speakers at time $t$ , with $y(t) = 1-x(t)$ for Y-speakers. More elaborate versions include a bilingual state with fraction $z(t)$ such that $x+y+z=1$ (Tamm et al., 12 Nov 2025).

The essential advance distinguishing threshold models from the Abrams-Strogatz (AS) framework is the incorporation of discrete, memory-dependent learning or shift thresholds. Let $K_X^*, K_Y^*$ denote the minimal number of encounters/exposures required to trigger a language shift toward X or Y, over a memory window $T^*$ . The associated critical speaker fractions are $x^* = K_X^*/(\mathrm{int} \, T^*)$ , $y^* = K_Y^*/(\mathrm{int} \, T^*)$ , introducing thresholds $x^*, y^* \in (0,1)$ below (or above) which transition rates are suppressed (Tamm et al., 14 Jun 2024).

Transition rates for language shift are then governed by step or sigmoid functions:

Sharp (step): $r_x(x) = j_x \Theta(x - x^*)$
Smooth: $r_x(x) = j_x H(x; x^*, w)$

where $\Theta$ is the Heaviside function, $H$ is a smoothed sigmoid, and $j_x$ is a maximal rate parameter.

These threshold-modified rates induce piecewise-linear or piecewise-constant evolution equations, segmenting the state space into regions defined by the relative placement of speaker fractions and thresholds.

2. Dynamical Regimes and Analytical Solutions

The partitioning of the state space yields distinct dynamical regimes, most economically captured in the two-language threshold model (Tamm et al., 14 Jun 2024). The time-evolution is

$\frac{dx}{d\tau} = \gamma \, \Theta(x - x^*) (1-x) - (1-\gamma) \, \Theta(1-x-y^*)\ x$

where $\gamma$ is a "prestige" or status parameter.

Classification by the position of $x$ with respect to the thresholds leads to:

Region	Active Shifts	Dynamics	Attractor
I: $x > \max(x^, 1-y^)$	Only Y→X enabled	$dx/d\tau = \gamma(1-x)$	$x \to 1$ (X consensus)
II: $x < \min(x^, 1-y^)$	Only X→Y enabled	$dx/d\tau = -(1-\gamma)x$	$x \to 0$ (Y consensus)
III: $x^* \leq x \leq 1-y^*$	Both shifts enabled	$dx/d\tau = \gamma - x$	$x \to \gamma$ (coexistence)
0: $1-y^* < x < x^*$	Both shifts suppressed	$dx/d\tau = 0$	frozen at $x(0)$

A key bifurcation condition for coexistence is $x^* + y^* < 1$ ; only then is the interval $[x^*, 1-y^*]$ non-empty.

Stability is determined by the sign of derivatives in each regime: all attractors above are globally attracting within their respective basins. No interior stable coexistence is possible under the original AS model for $a \ge 1$ power-law rates.

In smooth-threshold models (sigmoid $H$ replacing $\Theta$ ), the basic structure persists: three attractors (consensus on X, consensus on Y, stable coexistence) and a "frozen" or very slow-drifting band, with transitions between regimes shifted according to sigmoid width.

3. Empirical Validation and Predictive Power

Threshold models attain superior empirical fit over AS-type models in a range of observed language shift datasets. For six historical cases—including Welsh in Wales (1891-2021), Swedish in Finland (1880-2020), Quechua in Peru, French in Canada, English in Quebec, and Russian in Estonia—fitting the step-threshold model produces lower mean squared error (SSE) and faithfully reproduces observed slow-downs, plateaux, and abrupt regime shifts (Tamm et al., 14 Jun 2024).

Key findings include:

Welsh: threshold + coexistence regime matches stabilization at plateau (SSE $2.7 \times 10^{-4}$ vs. AS at $8.9 \times 10^{-4}$ ).
Scottish Quechua: two-stage decay replicated only by threshold model.
Swedish in Finland: nonzero coexistence equilibrium achieved only by threshold model.
In three quasi-frozen cases (Canada, Quebec, Estonia), threshold model explains extremely slow drift as consequences of sub-threshold fractions, wherein conventional AS predicts rapid extinction.

This supports the interpretation that memory, conscious learning decisions, or sociocultural thresholds are critical in language fate, not just continuous prestige gradients.

4. Structural Role of Thresholds and Bifurcations

Thresholds introduce discrete bifurcation surfaces into the phase diagram, fundamentally altering possible long-time outcomes. In step-threshold models, phase space is partitioned into regime domains by simple linear conditions (e.g., $x^* + y^* = 1$ ), each domain associated with a fixed-point attractor or manifold.

In smooth-threshold variants, bifurcation curves become continuous but the set of qualitative scenarios remains: extinction, coexistence, freezing. As thresholds are crossed, regimes can switch rapidly—e.g., from prolonged near-stasis to rapid shift once one language breaches its memory threshold.

Critical slowing down is observed as the system approaches threshold bifurcations, manifest in diverging convergence times. This provides an empirical early-warning signal: as control parameters (e.g., exposure, prestige) approach tipping values, change becomes increasingly sluggish before the switch between regimes.

5. Connections to Multi-Language and Bilingual Models

Threshold phenomena generalize to competition models with more than two languages, the inclusion of bilingual or intermediate states, or asymmetric status biases (Tamm et al., 12 Nov 2025, Zhou et al., 2019, Otero-Espinar et al., 2013). In the symmetric three-state threshold model (monolingual X, monolingual Y, bilingual), each of four step-threshold functions divides the population simplex into regions with distinct linear dynamics, giving rise to extinction, coexistence with/without bilingual mediation, and neutral frozen states.

In these extensions, learning and attrition thresholds tuned by bilingual prevalence can stabilize long-lived bilingual or trilingual equilibria, matching real-world persistence (e.g., Basque, Catalan, Welsh) which continuous-rate models cannot. Bifurcation manifolds and scaled threshold parameters $s_\pm$ classify the phase space, with critical lines (e.g., $s_+ = \bar s$ ) delimiting the onset of stable all-state coexistence.

6. Theoretical Implications and Comparative Perspective

Threshold models highlight the centrality of nonlinear, memory-mediated learning processes, as opposed to purely opportunistic language adoption proportional to group size or prestige:

Memory-limited exposure imposes hard cutoffs below which language shift is infeasible, even if prestige favors such a shift.
Threshold effects can create robust plateaux and "freezing" domains invisible to models based solely on continuous rates.
Empirical regimes of prolonged stasis, coexistence, or abrupt language collapse are naturally explained as instances of threshold-driven dynamics.

In contrast, original AS and related power-law models generally predict only bistability or monostable extinction, with no provision for neutral bands or stable interior coexistence unless volatility is sublinear ( $a<1$ ), a condition contradicted by most empirical fits (Seoane et al., 2017).

Presence of thresholds is further validated by contact-induced language change models, where L2-difficulty thresholds in the presence of proportion $\sigma$ of L2 speakers establish conditions for rapid loss or durable retention of complex grammatical features (Kauhanen, 2021).

7. Practical and Analytical Methodology

Parameter estimation in threshold models requires historical time-series (for $x(t)$ ), inference of critical fractions from sociolinguistic or demographic surveys ( $K_X^*, K_Y^*, T^*$ ), and fitting by nonlinear regression (minimizing SSE). In smooth-threshold cases, sigmoid sharpness $w$ can be inferred from population heterogeneity.

Threshold locations and regime boundaries in parameter space are determined analytically in piecewise-linear models, or numerically in multiple-state or smooth-threshold settings. The resulting phase diagrams partition parameter space into extinction, coexistence, and frozen regions, with analytic formulas available for attractor locations and separatrices.

Continuous monitoring of convergence times provides early warning of critical transitions; slowdowns predict imminent regime bifurcation and can guide intervention to preserve linguistic diversity (e.g., raising exposure or prestige above the critical threshold) (Zhou et al., 2019).

The threshold model, by incorporating explicit memory, learning, and social-pressure cutoffs, produces a taxonomy of language competition outcomes more closely aligned with observed multilingual persistence and shifts than preceding continuous-rate frameworks. Threshold-induced phase transitions underlie the dynamical repertoire seen in empirical case studies and provide rigorous tools for predicting, interpreting, and potentially steering language evolution in real populations.