Brief guide to use the applet

Here it is assumed that you are familiar with the dynamical rules of the models as described in the pdf presentation and references quoted above. Function buttons: start/stop button: begins and stops running the simulation. ms/frame: set the waiting time between two consecutive frames. A larger value of ms/frame runs the dynamics slower. + and - buttons: increase/decrease the basic input parameters of the model. Initial conditions buttons: random i.c.: language use (A red, B black or AB bilingual white) of each agent is randomly distributed. Set by default. strip-like i.c.: half of the lattice is using language A (red), while the other half is using language B (black). AB_init_domain: half of the lattice is bilingual (white), while the other half is monolingual A (red). INPUT parameters of the model: L: linear dimension of the lattice. Size of the system: N = L x L s (0<s<1): social prestige of language A (red). The social prestige of language B (black) is 1- s. The case s=0.5 is the case of socially equivalent languages. Language A is preferred for s>0.5. a (a>0): volatility or persistence parameter. The case a=1 is the marginal situation, where the transition probabilities depend linearly on the densities of speakers. A high volatility regime regime exists for a<1, with a probability of changing language state above the marginal case. A low volatility regime exists for a>1 with a probability of changing language state below the marginal case. **Note that you can change the parameters a and s during the dynamics with the corresponding buttons! OUTPUT. The instantaneous value of the following quantities are given: density_A, density_B, density_AB: densities of speakers of each linguistic group (A, B and AB). ρ: average interface density defined as the density of links joining two agents in a different state. The inverse of ρ gives an average linear measure of a monolinguistic domain. Decreasing ρ shows that linguistic domains are being formed and grow in time. When ρ=0 the dynamics reaches a final absorbing state with every agent using the same language. time: unit of time of the simulation. A unit of time includes N iterations of the dynamics so that each agent has been updated on average once every unit of time.

Exploring the parameter space

As a guide to explore the parameter space to observe different qualitative dynamical behavior you can use the following parameter values: **System size of N=64 (set by default) is the best size to understand the dynamics. Random initial conditions are set by default. Parameter space (a,s) **Note that it is interesting to change parameters during the time evolution of the dynamics. Specially moving from the default case (a=1, s=0.5) to the other multiple situations described below. 1) Marginal volatility: a=1 1.1) a=1, s=0.5: Case of socially equivalent languages. We observe formation and growth of monolingual domains. After a long time, a finite size fluctuation drives the system to consensus on one of the two languages and extinction of the other. In the Abrams-Strogatz model, the dynamics is driven by interfacial noise (Voter Model-like); while in the Bilinguals model dynamics is curvature driven and bilingual agents do not form domains but remain at the interfaces between monolingual ones. Starting the Bilinguals model from random initial conditions, 1/3 of the times the system gets trapped in stripe-like dynamical metastable states, where the system remains in dynamical coexistence for long times. **Other initial conditions: - strip-like i.c.: with this initial condition, the different interface dynamics of the two models can be clearly seen. In the Bilinguals model, the system gets trapped in stripe-like metastable states (due to surface tension), while in the Abrams-Strogatz model, boundaries diffuse by noisy interface dynamics, breaking the initial strip-like structure. - AB_init_domain (just in the Bilinguals model): with this initial condition, one can see how bilingual domains are not stable, but break into smaller monolingual domains. Bilingual agents play a role only at the interfaces. 1.2) a=1, s>0.5: (try s=0.6) Case where language A is more prestigious than language B. A fast decay of the less prestigious language is observed. 2) High volatility regime: a<1 2.1.1) a<1 (try a=0.4), s=0.5: Abrams Strogatz model becomes more noisy as a approaches zero. The Bilinguals model, looses progressively surface tension as a→0 (continuous transition), leading to less defined borders between monolingual domains. 2.1.2) a<1 (try a=0.4), s>0.5 (try s=0.6): We observe the extinction of the less prestigious language (i.e, B) in both models; but for slightly differences in prestige, i.e. s belonging to the interval [0.4,0.6], in the Bilinguals Model extinction is much faster. 2.2.1) a=0.2, s=0.5: At such small value of a both models behave very similar: noisy dynamics leading to long lived coexistence without formation of domains. No well defined linguistic borders. 2.2.2) a=0.2, s>0.5 (try s=0.6): However, the effect of s is different: while in the Abrams-Strogatz model there is no dramatic effect when s is in the interval [0.4,0.6] (still long lived coexistence), in the Bilinguals model even such small asymmetry in the prestige causes a fast extinction. 3) Low volatility regime: a>1 3.1) a>1 (try a=3.0), s=0.5: Abrams-Strogatz model gains progressively surface tension as a increases. In both models, and for a sufficiently large, during the coarsening process monolingual domains develope very flat boundaries due to the inertia to change language state (low volatility). Both models behave very similar at this stage. 3.2) a>1 (try a=3.0), s>0.5 (try s=0.6): Both models behave very similar: extinction of the less prestigious language. However, as a gets larger, the time to extinction increases. Generated by Xavi Castello