In order to investigate the impact of memory on language competition, we develop an agent-based model of a population of N speakers, of whom a fraction N _X speak language X, a fraction N _Y speak language Y, and a fraction N _Z are bilinguals speaking both languages X and Y; N _X + N _Y + N _Z = 1. In the following, the symbols X and Y also represent the language state of X- and Y-monolinguals, respectively, while Z represents the state of bilingual agents. It is assumed that initially, there are no bilinguals, i.e., N _Z (t = 0) = 0.

During the time evolution of the system, agents communicate pair-wise. Each time step in the simulations corresponds to 1 day, and an agent is assumed to communicate every day on average with other N _int agents. Thus, there are daily N _int × N/2 interactions in the system.

For the choice of the language used in a communication, we apply the following rules.

• In the encounter between two monolinguals of different languages, since there is no common language to be used, it is assumed that either speaker will try to use their own language, which the other speaker will make an effort to understand so that eventually, either agent will learn something of the other agent’s language.

In the learning process, memory is taken into account, assuming that a monolingual agent can acquire the other language and become bilingual only through repeated interactions with monolingual speakers of the other language. In the model, we assume that in order to learn a new language and become bilingual, a monolingual agent has to use the new language at least K times within a time interval T ^K . If the monolingual agent has used the new language less than K times, as time interval T ^K elapses, the agent’s memory is reset. Importantly, the model will use, in general, two different parameters K ^X and K ^Y for the two languages X and Y, respectively, reflecting the possible different difficulties of learning; i.e., K ^X (K ^Y) expresses the difficulty of learning language X (Y) and, therefore, will be used to determine if a monolingual agent speaking language Y (X) meets the criterion to become bilingual or not.

Memory effects also influence language maintenance. In the model, we assume that a bilingual has to use both languages X and Y at least M times within a time interval T ^M in order to maintain them (for simplicity, we assume the same M and T ^M for both the first and second languages). If this condition is not fulfilled for one of the languages, that language will be forgotten.

The model variables are discussed in Section 2.2, and the dynamical rules of time evolution are listed in Section 2.3.

The state of agent i (i = 1, 2, … , N) at time step t is specified by the following set of individual variables:

• Language variables (all agents): The linguistic state of agent i is recorded by the variable L i [ 1 ] , representing the (main) language ( L i [ 1 ] = X or Y), and, if the agent is bilingual, also by the variable L i [ 2 ] , recording the second language ( L i [ 2 ] = Y or X, respectively). For a monolingual agent, L i [ 2 ] = 0 .

The algorithm employed in the numerical simulations can be summarized as follows:

0. At time t = 0, the population contains only a fraction N _X (0) of X-monolinguals and a fraction N _Y (0) of Y-monolinguals, i.e., N _X (0) + N _Y (0) = 1 and N _Z (0) = 0. Counters τ i k and k _i are initialized to zero.

1. A set of N _int × N/2 couples of agents are randomly selected from the population.

2. Agents i and j of each extracted couple interact, and their language use is tracked (Table 1; Table 2).

• If both agents i and j are monolinguals of the same language ( L i [ 1 ] = L j [ 1 ] ) , nothing happens.

3. After the N _int × N/2 couples of agents extracted have interacted, the time variables are updated for each agent i: if agent i is monolingual, time counter τ i k is updated, τ i k → τ i k + 1 ; if agent i is bilingual, time counter τ i m is updated, τ i m → τ i m + 1 . Furthermore, the system time t is updated, t → t + 1.

4. Language use counters, time counters, and possible transitions between the monolingual and bilingual state are checked for each agent i (see Table 3).

• If agent i is monolingual, the counters k _i and τ i k are checked at each time step. If k _i < K and τ i k < T K , the learning process continues. If, instead, k _i < K and τ i k = T K , the counters are reset, k _i → 0 and τ i k → 0 , and the learning process starts over. If k _i ≥ K and τ i k ≤ T K , agent i becomes bilingual, and the maintenance variables, τ i m , m i [ 1 ] , and m i [ 2 ] , are initialized to zero.

5. The algorithm is reiterated from step 1 until the final simulation time t = t _fin is reached or until all speakers become monolinguals of the same language.

It should be noted that instead of using an algorithm where the time and language use counters are reset, as explained previously, one could use an alternative more realistic algorithm based on a moving temporal window, i.e., checking at each time step the language use during the reference time intervals T ^K or T ^M. For computational convenience, we used the aforementioned algorithm, but we checked that within the parameter range used, the two methods provide equivalent results.

As discussed in Introduction, in a language dynamics model that aims to scrutinize the impact of memory and learning processes, it is relevant to take into account that individual linguistic skills to learn or retain a language are different. This heterogeneity may depend on many factors. In our model, the effect of these factors is described in terms of a heterogeneous distribution of the parameters regulating the learning and maintenance of a language, i.e., the associated time intervals and language usage frequencies.

As far as the learning process is concerned, the global language usage thresholds K ^X and K ^Y for learning language X and Y by monolingual speakers of language Y and X, respectively, are replaced by a heterogeneous set of parameters K i X and K i Y , respectively, assigned to each agent i. Analogously, to include heterogeneity in language retention, in place of a single value T ^M, we extract a different value T i M for each agent i. Thus, the time period after which a speaker forgets a language is an individual characteristic.

In the following equation, we limit ourselves to considering a specific case with a simple correlation between the threshold values of language usage and attrition times of generic agent i. K i X = K X 1 + S i σ X , (1) K i Y = K Y 1 + S i σ Y , (2) T i M = T M 1 − S i σ M . (3)

In the aforementioned three equations, S _i is the same random number extracted from a standard normal distribution, the values K ^X, K ^Y, and T ^M represent the means, and σ ^X, σ ^Y, and σ ^M represent the relative standard deviations of the respective distributions. Thus, the standard deviations express the degree of heterogeneity of the agents. Equations 1, 2 describe the learning, and Eq. 3 describes the language maintenance features of individual i. The Gaussian distribution is the most natural one to describe the heterogeneity of the linguistic skills in a population. A value S _i > 0 in Eq. 1 or 2 describes a speaker who needs for learning to practice language X or Y more times than on average; instead, S _i < 0 corresponds to the opposite case. The negative sign in Eq. 3 reflects the fact that, in general, the individuals who learn more slowly are, at the same time, the ones who forget more quickly, and vice versa. For the sake of simplicity, we assume that σ ^X = σ ^Y = σ ^M ≡ σ.

It should be noted that since in Eqs 1–3 S _i is extracted from a standard normal distribution, K i X , K i Y ,and T i M can take also negative values. As long as σ is small enough, this happens only for a small fraction of the extracted values of S _i . In fact, in the following section, we limit ourselves to studying the parameter range σ ∈ (0, 0.5]. For σ = 0.5, agents corresponding to the upper end of the range defined by ± one standard deviation from the mean learn three times faster (and forget three times more slowly) than agents positioned at the opposite end. Larger values of σ would lead to unrealistic differences between the individuals. For a given value of σ, in the case an S _i value leads to a negative parameter K i X , K i Y , or T i M , that parameter is set to zero.

We simulate the model for a population comprising at least N = 10³ agents, unless specified differently. We checked that this is a sufficient population size and the results remain unchanged when increasing the number of speakers even up to N = 10⁴. The simulation time is N _t = 4 × 10⁴ days ≈ 110 years. We assume that each time step in the simulations corresponds to 1 day and each agent has an average number N _int of daily interactions with other agents. Based on psycho-sociological and epidemiological studies (Valle et al., 2007; Zhaoyang et al., 2018) about social interactions, we explore a range of N _int ∈ [8, 24]. We allow for monolinguals of language Y (X) a time T ^K = 730 days = 2 years to acquire language X (Y), during which they should use the language at least K ^X (K ^Y) times. When exploring the symmetrical case of an equal level of difficulty of language learning, we set K ^X = K ^Y = 5,110 (i.e., on average, seven practicing events per day to learn a new language). In the asymmetrical case, K ^Y is maintained at 5,110 and K ^X is varied. The ability of bilinguals to retain the native or acquired language is assessed over the time period T ^M = 7,300 days = 20 years, during which the speakers, to remain bilinguals, should use both languages at least M = 43,800 times (on average, six times per day). Unless specified differently, we set σ = 0.5 when diversifying K ^X, K ^Y, and T ^M in order to add heterogeneity to the aforementioned model.

It should be noted that the choice of the minimum number of learning interactions needed to acquire a new language, determined by the thresholds K ^X, K ^Y, and T ^K , as well as of the minimum number of practicing events required to retain a language, determined by M and T ^M, is arbitrary. However, studying the dependence of simulation results on the number of daily interactions N _int per agent is equivalent to varying the aforementioned thresholds while keeping N _int constant. This is because the competition outcome is mainly determined by the interplay between the daily interaction frequency and model thresholds.

We begin our study of the effects of memory on language competition dynamics by examining first the case of a homogeneous population.

As long as no bilinguals are present in the system (beginning of the time evolution), a speaker interacts daily, on average, N _X (0)N _int times with X-monolinguals and N _Y (0)N _int times with Y-monolinguals. As discussed in Section 2.3, a monolingual → bilingual transformation, X → Z (or Y → Z), takes place only if an X (or Y)-monolingual has used the other language at least K ^Y (or K ^X ) times within the maximum allowed learning time T ^K . This implies the following conditions for the transition times t _X and t _Y : t X ≡ K X / N i n t N X 0 ≤ T K f o r Y → Z , (4) t Y ≡ K Y / N i n t N Y 0 ≤ T K f o r X → Z . (5)

If both conditions are fulfilled, then which of the two possible transformations, Y → Z or X → Z, will take place, i.e., which monolingual community will form the bilingual population, depends on which of the two transition times, t _X ≤ T ^K or t _Y ≤ T ^K , is smaller. For given values of N _int and T ^K , this depends on K ^X, K ^Y, and the initial fractions N _X (0) and N _Y (0). In the symmetrical case, when K ^X = K ^Y ≡ K, the transition type is determined only by the values of N _X (0) and N _Y (0); if N _X (0) = N _Y (0), both monolingual populations turn into bilinguals; otherwise, it is the population of the minority language that becomes bilingual. In the asymmetrical case, when K ^X ≠ K ^Y, for equal initial population fractions, N _X (0) = N _Y (0), the transition type is determined by the values of K ^X and K ^Y: for K ^X > K ^Y, the bilingual community will be formed by X-monolinguals and for K ^Y > K ^X by Y-monolinguals, i.e., by the monolinguals of the language that is more difficult to learn. If neither condition is fulfilled, no bilingual community will appear.

In order to illustrate the beginning of the system time evolution, let us consider the example of a population of 100 identical speakers (we use in this example only 100 individuals for the clarity of visualization) with initial population fractions N _X (0) = 0.6 and N _Y (0) = 0.4. The daily interaction rate is N _int = 16, K ^X = K ^Y = 5,110, and the transition times for the given parameters are t _X = 532 and t _Y = 798, respectively. In this case, since t _X < T ^K < t _Y , all the Y-monolinguals will become bilinguals around time t = t _X . Accordingly, in Figure 1A, the lines depicting the usage counters k _i of the Y-monolinguals stop, while the lines corresponding to the bilinguals’ language usage counters m i [ 1 ] and m i [ 2 ] appear; because N _X (t) > N _Y (t) + N _Z (t) = N _Y (0), the lines corresponding to m i [ 2 ] ( t ) measuring the usage of the acquired language X have larger slopes than the lines corresponding to m i [ 1 ] ( t ) measuring the usage of the bilinguals’ first language Y. Instead, because after time t _X , the monolinguals of language X will use only language X, both in the communication among themselves and with the bilinguals, their language usage counters k _i will remain constant until t = T ^K , when they are reset to zero.

Figure 1B shows the corresponding time evolution of the population fractions: at time t _X , there is the transition Y → Z, i.e., all monolinguals of language Y learn language X and become bilinguals; instead, N _X (t) = N _X (0).

Now, if the parameters are such that one of the monolingual populations becomes bilingual, whether this state is stable or not, it depends on the parameters regulating the attrition process. It will be stable if bilinguals practice both languages at least M times during a time interval T ^M. Because in the current example in the communication between bilinguals, only their first language is used (all bilinguals have the same first language), then N _Z (t) = N _Y (0) (as in our example depicted in Figure 1) or N _Z (t) = N _X (0); the second language is used by bilinguals only in the communication with monolinguals, whose population is equal to the initial one. Consequently, the conditions for the bilingual population to maintain languages X and Y are, respectively, T M N i n t N X 0 ≥ M to maintain language X , (6) T M N i n t N Y 0 ≥ M to maintain language Y . (7)

The bilingual community remains bilingual only if both conditions are satisfied. If one of the conditions is not fulfilled, the respective language is forgotten and the bilinguals become monolinguals of the language for which the condition is satisfied. If none of the conditions is fulfilled, the bilinguals become monolinguals of their original language. In the aforementioned example (Figures 1A, B), the bilinguals maintain both languages and the state reached is stable.

The behavior of the system under different initial conditions is illustrated by the phase portraits in Figure 2A for K ^X = K ^Y (symmetrical language learning difficulty) and Figure 2B for K ^X ≠ K ^Y (asymmetrical language learning difficulty). Possible initial conditions are represented by the points on the line N _X (0) + N _Y (0) = 1, where N _Z (0) = 0. In this example, we chose N _int = 10, allowing us to observe different regimes: some trajectories evolve toward a final state, where, in addition to bilingual speakers, there are also monolinguals of language X (or Y)—as shown by the dots on the N _X -axis (N _Y -axis); for the rest of the initial conditions, the representative point remains in the initial state. This behavior can be understood by expressing conditions (4)–(5) in terms of the initial fractions N _X (0) and N _Y (0). N X 0 ≥ K X / N i n t T K ≡ N X * f o r Y → Z , (8) N Y 0 ≥ K Y / N i n t T K ≡ N Y * f o r X → Z , (9) which define implicitly the critical initial fractions N X * and N Y * , above which a transition to bilingualism takes place. In the symmetrical case, when K ^X = K ^Y = 5,110, the values of the critical initial fractions are N X * = N Y * = 0.7 . For the asymmetrical case with K ^X = 5,840 and K ^Y = 4380 depicted in Figure 2B, N X * = 0.8 and N Y * = 0.6 . No transition to bilinguals takes place if the trajectory starts from an initial condition for which neither inequality is satisfied, i.e., when N X ( 0 ) < N X * and N Y ( 0 ) < N Y * , corresponding to the interval 1 − N Y * < N X ( 0 ) < N X * (see Figures 2A, B). In this case, the two monolingual communities remain isolated and coexist together so that N _X (t) = N _X (0) and N _Y (t) = N _Y (0). If the parameters are such that 1 − N Y * > N X * , such an interval does not exist; for example, for the K ^X, K ^Y, and T ^K values used in Figure 2, it disappears in both symmetrical and asymmetrical cases, if N _int ≥ 14.

Now, from the last remark, as well as from conditions (4)–(5) and (6)–(7), it is also clear that the number of daily interactions, N _int, is a very important quantity in determining the dynamics and outcome of the language competition. In Figure 3, the final population composition is plotted as a function of N _int, dashed lines, for the homogeneous system. This figure shows clearly how the final outcome of the language competition is determined by the interplay of the learning and attrition processes. Let us determine the conditions for the learning and maintenance of languages X and Y for the example with K ^X = K ^Y = 5,110 and N _X (0)/N _Y (0) = 60/40. From Eqs (4) and (5), one obtains the conditions on the interaction rate: N _int ≥ K ^X /N _X (0)T ^K = 11.6 and N _int ≥ K ^Y /N _Y (0)T ^K = 17.5, for X- and Y-monolinguals, respectively. Therefore, nothing happens until the value N _int = 11.6 is reached (it should be noted that in Figure 3, N _int assumes only integer values), and the two populations speaking languages X and Y remain isolated and do not generate any bilingual community. At N _int ≥ 11.6, Y-monolinguals learn language X and become bilingual. However, in order to remain as such, they have to maintain both languages. According to Eqs (6) and (7), the condition for a bilingual to maintain language X is N _int > M/N _X (0)T ^M = 10, whereas for maintaining language Y, the interaction frequency has to satisfy the condition N _int > M/N _Y (0)T ^M = 15. Thus, in the interval 11.6 ≤ N _int ≤ 15, bilinguals who appeared in the system during the time evolution will turn into X-monolinguals so that there are only X-monolinguals in the final state of the system. Instead, for N _int > 15, the bilinguals can maintain both languages, and the final population consists of X-monolinguals and bilinguals. Thus, depending on the value of the interaction frequency, one can observe three regimes (see the dashed lines in Figure 3): 1) coexistence of the two monolingual populations (with N _X (t) = N _X (0) and N _Y (t) = N _Y (0)); 2) only the majority language surviving; and 3) coexistence of monolingual speakers of majority language and bilinguals whose first language is the minority language.

Not taking the memory effects into account, i.e., assuming a time unit much larger than the time scales T ^K and T ^M, the homogeneous version of the proposed model, discussed in the this section, reduces to the Minett–Wang model that is local in time. The Minett–Wang model with neutral volatility has only two stable equilibrium points: one of the two competing languages surviving and the other one disappearing; which of the two languages survives depends on the model parameters. Instead, introducing the time-consuming language learning and retention processes, for a wide range of parameters, both languages can survive due to two opposing mechanisms: 1) not learning the other language (coexistence of the two isolated languages) or 2) frequent language contact between the minority language speakers, who eventually become bilinguals, and monolinguals of the majority language.

In the example illustrating the time evolution in a homogeneous population—Figures 1A, B—all monolinguals of language Y become bilinguals around time t _X . The population time evolution is characterized by a simple behavior and sharp transitions because agents are identical to each other in their features and they behave in the same way. The only source of randomness in the homogeneous system is related to the random choice of the interacting agents.

In the corresponding example of a heterogeneous population—Figures 1C, D—because in conditions (4)–(5), K ^X and K ^Y are replaced by K i X and K i Y , respectively, for each individual i, the transition Y → Z takes place at a different time, resulting in a bundle of parallel lines, as shown in Figure 1C, instead of the overlapping ones, as in (A).

As a consequence, the step-like curves describing the transitions N _Y → 0 and N _Z → 0.4 are replaced by smoother sigmoids. Depending on the value of σ, it can also happen that not all monolinguals of language Y become bilingual at some time t ≤ T ^K , whereas some X-monolinguals may do so. Similar considerations are valid for the attrition. In fact, in Figure 1C, one can notice that there is a line m i [ 1 ] ( t ) which has a slope much larger than that of the other lines, and a line m i [ 2 ] ( t ) with slope equal to zero. These lines correspond to a bilingual that used to be an X-monolingual and not Y-monolingual as the other bilinguals in this figure.

Such diversity in the behavior of individuals induces significant differences in the dynamics and final competition outcome between the homogeneous and heterogeneous systems. To understand how this happens, let us have a look at the time evolution of the speakers’ fractions N _X , N _Y , and N _Z on a longer time scale, depicted in Figure 4, for a heterogeneous population with σ = 0.5, initial ratio N _X (0)/N _Y (0) = 60/40, K ^X = K ^Y = 5,110, and three different values of N _int corresponding to low (N _int = 8), medium (N _int = 16), and high (N _int = 24) daily interaction rates.

In the example of the homogeneous system (Figures 1A, B), only the speakers of minority language Y contributed to the formation of the bilingual community. Instead, in the heterogeneous system, as shown already in Figure 1C, the speakers of majority language X also contribute (see the initial decrease in N _X in all panels of Figure 4), but their contribution is smaller than that of Y-monolinguals. It should be noted that the larger the interaction rate, the larger the contribution of Y-monolinguals (compare the three panels in Figure 4). In the following, the formed bilingual community decreases in time due to the attrition process, leading to a corresponding growth of N _X . The decrease in N _Z and the corresponding increase in N _X become larger for higher values of N _int.

The appearance of a maximum in the bilingual population size is also observed in the Minett–Wang model (Minett and Wang, 2008; Heinsalu et al., 2014), in which, however, finally only one of the languages survives, as mentioned previously.

Before proceeding with the discussion, we analyze the presence of the small zigzagged segments in the time evolution curves shown in Figure 4. They are related to attrition processes taking place at times close to each other due to diversity and initial conditions. We verified that they tend to disappear for t → ∞ and their amplitudes decrease with the system size, as N → ∞. We also checked that they do not significantly influence the final results.

Figures 4A, B show that in a heterogeneous system, it is possible for the two monolingual and bilingual populations to coexist in the final state. The same becomes evident from the phase portraits in Figures 2C, D and from the curves shown in Figure 3 for the heterogeneous system. Although in Figure 3, the dashed curves for the homogeneous system and the solid ones for the heterogeneous system might seem to be qualitatively similar, the fraction of Y-monolinguals decreases and the fraction of bilinguals increases with N _int, and the fraction of X-monolinguals passes through a maximum, and even the phase portraits for heterogeneous and homogeneous populations in Figure 2 might seem to be not so different, the systems are actually fundamentally different. That is, it appears that adding heterogeneity to the population can, for a wide range of parameters, lead to the coexistence of the three linguistic groups, which is not possible in a corresponding homogeneous system. Therefore, we conclude that diversity in linguistic abilities is one of the key factors leading to the possibility of the coexistence of all linguistic groups in a three-state system, and this is the most important difference between the homogeneous and heterogeneous systems.

The reason behind the smoothing of the curves shown in Figure 3 when adding heterogeneity in the population is similar to why the step-like transitions shown in Figures 1B are replaced by smoother curves shown in Figures 1D. That is, the replacement of K ^X, K ^Y, and T ^M by K i X , K i Y , and T i M consequently satisfies conditions (4)–(5) and (6)–(7) for each individual i at a different value of N _int, instead of a global fixed N _int value, as in the case of a homogeneous population.

In order to evaluate how the degree of heterogeneity, expressed as the relative standard deviation σ, influences the results, we ran simulations for the same system with N _X (0)/N _Y (0) = 60/40 and K ^X = K ^Y = 5,110 studied previously, varying σ between 0 and 0.5. The results are depicted in Figure 5 for three different values of N _int. Furthermore, the figure shows that adding a certain level of heterogeneity to the system can result in a situation where, in the final state, all three linguistic groups are present in the system: see the lowest group of lines appearing as σ increases. For fixed parameter values and initial composition of the total population, the value of σ leading to such a final state increases slightly when N _int is increased. Furthermore, the diversity of individuals increases the competition advantage of the majority language, with respect to the homogeneous system, for all values of N _int (c.f. also Figure 3), and the effect of varying σ is the largest at intermediate values of N _int.

From the discussion in Section 3.2, it became clear that the initial conditions play a relevant role in determining the final outcome of the language competition in the homogeneous system. This is also true in the case of heterogeneous populations, as observed from Figures 2C, D and from Figure 6. In the case of equal initial population sizes, N _X (0) = N _Y (0) = 0.5, equal fractions of X- and Y-monolinguals become bilingual. If the interaction frequency is low, e.g., N _int = 8, at equilibrium, relatively large X- and Y-monolingual populations coexist with a small bilingual community. The size of the formed bilingual population increases as the interaction frequency N _int increases; Figure 6 shows the results for N _X (0)/N _Y (0) = 50/50 for three different values of N _int. If the ratio N _X (0)/N _Y (0) is increased, i.e., X becomes the majority language and Y becomes the minority language, the final population composition tends to be more and more dominated by majority language X, as shown in Figure 6. For N _X (0)/N _Y (0) ≳ 80/20, minority language Y becomes extinct, unless the interaction frequency is very low (N _int = 8). In the low interaction limit, the two language populations remain practically isolated from one another and can, in this way, coexist.

So far, we mostly concentrated on the symmetrical case K ^X = K ^Y ≡ K. Next, we examine in detail the asymmetrical case when K ^X ≠ K ^Y, i.e., the two competing languages have different degrees of learning difficulty.

From conditions (8)–(9) for the homogeneous population, it is clear that the asymmetry between the parameters K ^X and K ^Y induces asymmetry also in the critical fractions N X * , N Y * , e.g., for K ^X > K ^Y, also N X * > N Y * (c.f. Figures 2A, B). Therefore, the asymmetry in the language learning difficulty implies that in order to preserve a monolingual community of the language that is more difficult to learn and to induce, at the same time, the transition of the other (minority) language speakers into bilinguals, its minimal initial population fraction has to be larger with respect to the symmetrical case. Similar trends are also present for the heterogeneous systems (c.f. Figures 2C, D). In practice, this means that the more difficult a language is for the learners, the harder it is to integrate the foreign language community members, e.g., immigrants, and the smaller should their proportion in the population be, or/and some measures should be taken to favor the learning of the local language in order to avoid the formation of linguistically isolated communities.

Figure 7 presents the results for a heterogeneous system characterized by the parameter values summarized in Section 3.1 and with N _X (0) = N _Y (0) = 0.5. In this figure, the value of K ^X is increased from K ^X = K ^Y (corresponding to the symmetrical case) up to K ^X = 6K ^Y in order to observe how the asymmetry in the language learning difficulty influences the language competition outcome. As observed already in Figure 6, for N _X (0) = N _Y (0) and K ^X = K ^Y, in the final state, N _X = N _Y and the fraction of bilinguals increases with increasing values of N _int. Now, when K ^X > K ^Y (see Figure 7), the final fractions of X-monolinguals and bilinguals are decreased with respect to the symmetrical case, while N _Y (t _fin) is increased; the larger the N _int is, the larger the influence of asymmetry. However, increasing the asymmetry between the language learning difficulties beyond K ^X/K ^Y ≈ 3 does not substantially influence the results. For K ^X/K ^Y = 3, in order to become bilingual within the time interval T ^K , a speaker of language Y should practice language X, on average, 21 times per day. Comparing this with the number of average daily interactions (N _int ∈ [8, 24]), it is obvious that the task is very hard for any agent. Therefore, the monolinguals of language Y will remain monolinguals of their mother tongue and N _Y (t _fin) ≈ N _Y (0) = 0.5 for all studied values of N _int. Instead, the monolinguals of language X will become bilingual, the final fraction of bilinguals being larger for larger N _int values. In fact, for N _int = 24, almost all monolinguals of language X will turn into bilinguals, leaving only a very small fraction of X-monolinguals in the system.

Figure 8 illustrates the case where the initial populations have different sizes and set N _X (0)/N _Y (0) = 60/40. However, we now vary the value of K ^X so that the ratio K ^X/K ^Y goes from 1/7 to 4 (i.e., we go from the case where minority language Y is seven times more difficult to learn for monolinguals of language X than majority language X for monolinguals of Y, to the symmetrical case corresponding to K ^X/K ^Y = 1, up to the case where majority language X is three times more difficult to learn for monolinguals of Y than minority language Y for monolinguals of X). Figure 8 shows that for all studied values of N _int, an increase in K ^X/K ^Y, i.e., the learning difficulty of language X, leads to a decrease in N _X , as expected, and an increase in N _Y and N _Z . It should be noted that while in the case N _X (0)/N _Y (0) = 50/50, an increase of K ^X/K ^Y leads to a decrease in bilinguals (see Figure 7), in the case N _X (0)/N _Y (0) = 60/40, the fraction of bilinguals, instead, increases.