Model for phonetic changes driven by social interactions

We propose a stochastic model to study phonetic changes as an evolutionary process driven by social interactions between two groups of individuals with different phonological systems. Particularly, we focus on the changes in the place of articulation, inspired by the drift $/\mathrm{\Phi }/\to /\mathrm{h}/$ observed in some words of Latin root in the Castilian language. In the model, each agent is characterized by a variable of three states, representing the place of articulation used during speech production. In this frame, we propose stochastic rules of interactions among agents which lead to phonetic imitation and consequently to changes in the articulation place. Based on this, we mathematically formalize the model as a problem of population dynamics, derive the equations of evolution in the mean-field approximation, and study the emergence of three nontrivial global states, which can be linked to the pattern of phonetic changes observed in the language of Castile and in other Romance languages.

Figure 1

Stability analysis. Real part of the largest nontrivial eigenvalue of the Jacobian matrix (${\lambda }_1$), as a function of the parameter $Q$. (a)–(d) Numerical calculation for the equilibria I, II, III, and IV, respectively. Note that when $\text{Re}\left({\lambda }_1\right)>0$ the system is unstable; therefore, equilibrium I will be unstable for all values of $Q$; equilibria II and III will be unstable for $Q<2$, and equilibrium IV for $Q>2$.

Figure 2

Projection of the phase diagram in the plane $(n_A,Q)$. The curves $n_A=2/(2+Q)$ and $n_A=Q/(2+Q)$ divide the plane into four regions with different stability. The labels for each region correspond to the equilibria that are stable inside the region. The two blue markers represent the parameters that were used in the main text.

Figure 3

The case when equilibrium IV is stable. For $Q=0.7$, first moment evolution of the numbers related to (a) group A ($a_1,a_2,a_3$) and (b) group B ($b_1,b_2,b_3$), respectively. The curves are the outcome of the evolution of Eqs. (9), which were solved by performing a Runge Kutta eighth-order method. (c), (d) Average curves over 100 realizations, of the numbers in the ABM. The shading indicates the standard deviation around the mean value. In these conditions, the population evolves to equilibrium IV, a mixed state of pronunciation where the system reaches ${\vec{n}}_{\mathrm{eq}}=(\frac{5}{27},\frac{5}{27},\frac{7}{54};\frac{7}{54},\frac{5}{27},\frac{5}{27})\approx (0.18,0.18,0.13;0.13,0.18,0.18)$.

Figure 4

The case where equilibrium II is stable. For $Q=5$, the first moment evolution of the numbers related to (a) group A and (b) group B, respectively. (c), (d) The average curves over 100 realizations, of the numbers in the ABM. The shading in the plots indicates the standard deviation, which we see increasing when the system reaches the unstable equilibrium IV. Note how at the beginning the system seems to evolve toward IV, but it escapes toward equilibrium II, which is stable in these conditions [$\overrightarrow{n_{\mathrm{eq}}}=(0,0,\frac{1}{2};0,\frac{1}{4},\frac{1}{4})$]. The arrows in (a) indicate the time scales obtained from the eigenvalues analysis: $t_1\approx 1$ and $t_2\approx {10}^2$.

Figure 5

Average curves for the evolution of the numbers (a) $a_3$ and (b) $b_1$, for different values of the population size $N$, compared with the mean-field approximation in order to see the finite size effect. (c) Distribution of final times $T$ needed for the system to reach equilibrium II at $Q=5$. The dashed line in the background indicates a Gaussian distribution with the same mean and standard deviation, which helps to visualize the presence of a large tail in distribution $P\left(T\right)$. (d) Average curves for the evolution of the numbers $a_3$ and $b_1$ for different values of the parameter $r$.

Figure 6

Trajectories in the plane $a_1-a_3$: (a) changing the initial conditions, keeping $r=0.5$, and (b) changing parameter $r$, for trajectory starting from the same initial condition as $l_2$ in (a). The blue lines correspond to the solution of the differential equations, while the cyan curve in (a) corresponds to a single realization of the stochastic simulation, starting from the initial condition of trajectory $l_2$.