Dynamic Analysis of a Two-Language Competitive Model with Control Strategies

1 College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China 2Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea 3 Departamento de Análisis Matemático and Instituto de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain 4Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia


Introduction
The diversity of cultures is the greatest charm of the human civilization, and languages are the most important carrier for culture.In the past decades, with the progress of the globalization, local tongues are increasingly replaced by hegemonic languages [1], this trend that has been investigated from multiple points of view, including that of physics.We refer to some of them in [2][3][4][5][6][7][8][9][10][11] and the references therein.
Perhaps the earliest and simplest mathematical model for languages shift was developed by Abrams and Strogatz [2], Patriarca et al. [12,13], and Stauffer et al. [14].They considered a stable population in which two languages with different statuses competed for speakers and predicted that one of the languages would inevitably die out.The theoretical results were successfully fitted to historical data on the competition between Scottish Gaelic and English, Welsh and English, and Quechua and Spanish, among other language pairings [2].However, there was no mention of the fact that the possibility of bilingual individuals might exist, a possibility that is of course realized in numerous multilingual societies.This is widely exists in all parts of the world.For example, in Spain, Castilian Spanish is the official language throughout the state, but in certain regions it is coofficial with another language (mainly Galician, Basque, Catalan, or Valencian); individual bilingualism is common in communities with more than one official language.
Recently, Mira et al. [8,15,16] proposed a modified Abrams-Strogatz model that allows for bilingual as well as monolingual speakers of the competing languages and that includes a parameter that represents the ease of bilingualism.The model is accordingly described by the following differential equations: d d =   +   −  (  +   ) , d d =   +   −  (  +   ) , where capital letters  and  denote the two languages spoken in population; the uppercased letter  denotes the group of bilingual speakers; and the lowercased letters , , and  (with  +  +  = 1) refer to the fraction of speakers of each of the languages in population and the fraction of bilingual speakers, respectively; parameter   denotes the probability of a monolingual speaker of language  being replaced in the population by a monolingual speaker of language , with analogous notation for the other parameters   ,   ,   ,   , and   .The probability of a monolingual person being replaced by mono-or bilingual speaker of the other language is assumed to be proportional both to the status of the second language, that is, the social and/or economic advantages it offers, and to a power of the proportion of population that speaks it.Thus, denoting by  the relative status of language  and by 1 −  that of language , where  is a normalization factor related to the time scale,  is the power parameter, and  is the probability that the disappearance of a monolingual speaker of language  (resp., ) will be compensated for by the appearance of a bilingual rather than by a monolingual speaker of language  (resp., ).On basis of detailed analysis and extensive calculations, authors showed that both languages may coexist and survive in the long term.They pointed out that it is possible only if the competing languages are sufficiently similar, in which case its occurrence is favored by both similarity and status symmetry.
It is generally known that the disappearance of race language will bring the disappearance of race culture, even the whole disappearance of the corresponding race.The protection of endangered language has been concerned increasingly interdisciplinary in different contexts.
Very recently, the dynamical model with optimal control strategies has become a major topic in mathematical biology (see [17][18][19][20][21][22][23] and the references therein).Particularly, Joshi, in [18], proposed an HIV immunology model with optimal drug treatment strategies, and the existence and uniqueness results for the optimal control pair are established.Jung et al. [24] proposed a two-strain tuberculosis model with two control terms, and the optimal controls are characterized in terms of the optimality system.In addition, the state-dependent impulsive feedback control measure is also applied widely to the control of spread of infectious disease due to its economic, high-efficiency, and feasibility nature; see [25,26] and the references therein.
Motivated by these facts, in this paper, the dynamic behavior of two-language competitive model ( 1) with ( 2) is analyzed systemically in Section 2. A set of necessary conditions that an optimal control and state must satisfy, are derived in Section 3. In Section 4, we extend model (1) with state-dependent pulse control measure.Some sufficient conditions are presented in this section for the existence and stability of positive periodic solution.Some concluding remarks are presented in Section 5. (1) Inserting the parameters formula (2) into model ( 1) and taking into account  +  +  = 1, we get the following reduced model (here, we assume that  = 1 throughout the rest of this paper):
The following theorem is on the nonnegativity of solution of model (3).
To summarize the above discussion, we give some sufficient conditions for the existence and asymptotical stability of equilibria for model (3).Theorem 2. For any ,  ∈ (0, 1), one of the following statements is valid.Finally, we fix all parameters including , , , and  and carry out numerical investigations to confirm our main results obtained in this section.Firstly, we choose  = 0.5,  = 0.7,  = 0.5, and  = 1; it is easy to calculate that 1 −  = 0.3 < (1 − ) = 0.7 × 0.5 = 0.35.So, from the first conclusion of Theorem 2, we know that model (1) has only two trivial equilibria (1, 0, 0) and (0, 1, 0), (1, 0, 0) is a globally asymptotically stable node, and (0, 1, 0) is a saddle point, which is shown in Figure 1(a).It is clear that language  is permanent and language  will fade away in this case.Similar conclusion can be obtained from Figure 1(b) with parameters  = 0.3,  = 0.4,  = 0.5, and  = 1.However, if we choose  =  =  = 0.5 and  = 1, then languages  and  are coexistent and tend to positive equilibrium as shown in Figure 1(c).

The Protection of an Endangered Language by a Continuous Control Strategy
With the development of human civilization, people have taken effective measures to prevent the disappearance of language.In this section and the following, therefore, model (1) with parameters (2) is extended to assess the impact of control measures.And, in general, control strategies are divided into two main types: continuous control and pulse control.We, firstly, consider how a continuous control measure affects the dynamical behavior of model (1) in this section.Through discussion of Theorem 2 in Section 2, we know that (i) language  is permanent and language  and bilingual speakers  are extinct for (1 − ) < (1 − ) and that (ii) language  is permanent and language  and bilingual speakers  are extinct for  < (1 − )(1 − ).Considering the similarities of the two cases, we only need to consider case (i); that is, (1 − ) < (1 − ).and continuous control measure.The control system is modeled by the following differential equations:

Protecting of Language
where  is a controlled variable, which means that the fraction of language  becomes the bilingual  per unit of time.Note that  +  +  = 1; model ( 10) can be written as follows: We denote the right-side of model ( 11) by (, ) and (, ), respectively.We, here, discuss the existence and asymptotical stability of positive equilibrium of model (11).
Remark 5. From Theorem 2, it follows that language  is permanent and that language  and bilingual speakers  will eventually disappear for (1 − ) < (1 − ) in model (1).However, if we introduce a control variable  (no matter how small it is) in model ( 1), languages  and  are coexistent and tend to positive equilibrium ( *  ,  *  , 1 −  *  −  *  ).The coexistent state, of course, depends upon the controlled strength .This implies that  is a sensitive controlled parameter for the protection of endangered language.

Analysis of Optimal Control.
Optimal control techniques are of great use in developing the optimal strategies to protect endangered civilization.To solve the challenges of obtaining an optimal control measure, we use optimal control theory; for more details, see Lenhart and Workman [27].In model (10), for the optimal control problem, we consider a control variable () ∈   ; here,   = { : () is measurable, 0 ≤ () ≤ 0.9 for all  ∈ [0, final ]} indicates an admissible control.In this optimal problem, we assume a restriction on the control variable () such that 0 ≤ () ≤ 0.9, because conversion of all of language  at one time is impossible.In case of no control, the fraction of language  increases while the fractions of language  and bilingual  die out.Therefore, the biological meaning of an optimal control in this problem is that the adequate levels for the fractions of language  and bilingual  are built.Now, we consider an optimal control problem to maximize the objective functional subject to model (10).The first term represents the benefit of bilingual , and the other term is systemic cost of control measure.The positive constants  1 and  2 balance the size of the terms  and .Our goal is maximizing the fraction of bilingual  and minimizing the systemic cost to the control measure.This seeks an optimal control  * such that subject to model (10).It is obvious that the integrand of objective functional  is a convex function of control variable  and that state model satisfies the Lipschitz property with respect to the state since state solutions are bounded.The existence of an optimal control follows [28].
The necessary conditions that an optimal must satisfy come from the Pontryagin's Maximum Principle in [28].This principle converts optimal control problem (10) and ( 18) into a problem of maximizing pointwise a Hamiltonian  with respect to  as follows: where  1 ,  2 , and  3 are adjoint variables.
In the following theorem, we derive the necessary conditions for the optimal control problem.Theorem 6.Let ( * ,  * ,  * ) be an optimal state solution with associated optimal control variable  * for the maximized object functional () subject to control model (10).Then, for model (10), there exist adjoint variables  1 ,  2 , and  3 such that with transversality conditions   (  ) = 0,  = 1, 2, 3. Furthermore, the optimal control is given by Proof.From Pontryagin's Maximum Principle, adjoint variables   ( = 1, 2, 3) can be written as These are just differential equations ( 21)-( 23) with transversality conditions   ( final ) = 0 ( = 1, 2, 3).Furthermore, by optimality condition, we have that This shows that Using the property of control space, we get that This can be rewritten in compact notation, which is just (24).This completes the proof.

Numerical Simulation and Discussion
. Here, we discuss how the continuous control measure affects the protection of endangered language and the existence and stability of positive equilibrium for model (10) by numerical simulations.Firstly, we choose the same parameters as in Figure 1(a); that is,  =  = 0.5 and  = 0.7.Besides that, we choose control variable  = 0.25.Figure 2(a), however, is completely different from Figure 1(a), which shows that model (10) has a globally asymptotically stable positive equilibrium ( *  ,  *  ,  *  ) = (0.4194, 0.1244, 0.4562).Furthermore, from the discussion of Theorem 4, it follows that values  *  and  *  will decrease and increase with increase in control strength , respectively, which is shown in Figures 2(b) and 2(c).The strong consistency between theoretical result and real situation is obviously observed.
In addition, in Figure 2(c), it is interesting to note that control strength  is close to 1 (where  = 0.95), but language  runs around 0.2.This means that, no matter how strong the control strength  is, language  will not fade.Namely, control measure can only protect endangered language , but not result in extinction of language .Actually, in the real world, the own characteristics are very important factors in determining the development of languages.
The plots in Figure 3(a) show three adjoint variables  1 ,  2 , and  3 in the optimality system.We solve these adjoint equations by a backward Runge-Kutta fourth-order procedure because of the transversality conditions for more details, see Lenhart and Workman [27].In Figure 3(b), dotted line and solid line represent languages ,  and bilingual  in model (10) without and with continuous control, respectively.We see that the fractions of language  and bilingual  in population decrease more when there is no control.In this case, most of this population goes to language .If we apply the continuous control measure, however, the fraction of language  slowly falls, the fraction of language  decreases quite a lot, and the fraction of bilingual  is quite greater than the fraction in the case without control, since our main object is maximizing the fraction of bilingual .In Figure 3(c), the control  is plotted as a function of time for three different values of weight factor  2 : 0.045, 0.1, and 0.5.The control variable  for the associated weight factor  = 0.045 is much larger than the other two values.Note that, in general, as  2 decreases, the amount of  increases.The same results can also be obtained from the expression of  in (27).The associated weight factor  2 also plays a significant role in keeping the balance of the size of fraction an optimal problem.

Protection of Language 𝑌 with Impulsive
Control for (1−) < (1−) For reasons of protecting culture diversity, in this section, we will consider how the state-dependent pulse control measure affects the prevention of endangered language.Since learning cycle is very brief in contrast to the life cycle of a person, naturally we suppose that the procedure of learning is pulse effect.
As for the protection of endangered languages for the state-dependent control measure, we construct the following controlled model which is modeled by differential equations with state-dependent pulse effect: d d =   +   −  (  +   ) , The meaning of model ( 29) is as follows: when the fraction of language  reaches the critical threshold value   at time   , controlling measure (for example, encouraging some speaker of language  to study language ) is taken and the fractions of language  and bilingual speakers  immediately become (1 − )(  ) and (  ) + (  ), respectively.Remark 7. It is obvious that the fractions of language  and bilingual  are rather small and in danger of becoming extinct when the fraction of language  reaches the critical threshold value   .In this case, the effective measure is taken to prevent the loss of language .The times of control measures are obviously related to the state of language .Remark 8.The critical threshold value   represents the fraction of monolingual speakers  in population, and  represents the strength of control measure.Numerical simulations in Section 4.3 show that these are crucial parameters in model (29).
For model (29) Figure 3: The optimal adjoint variables, states, and control variable for the optimal control problem with  =  = 0.5,  = 0.7,  = 1, and  1 = 0.1: (a) the optimal adjoint variables  1 ,  2 , and  3 ; (b) the optimal states of languages ,  and bilingual ; and (c) the control variable  with different weight factor  2 .
Let S be an arbitrary set in R 2 , and let  be an arbitrary point in R 2 .The distance between point  and set S is defined by (, S) = inf  0 ∈S | −  0 |.For the convenience of statement in the rest of this paper, we introduce some definitions.

Main Results.
On the existence of positive order-1 periodic solution for model (30), we have the following theorem.
Remark 15.From Theorem 14, though condition (39) of Theorem 14 is hard to test, yet it is weak since the second and third items of the exponent term of the right-side of (39) are negative.
Next, we give a more general result on the existence and stability of positive order-1 periodic solutions of model (30).
Remark 17.According to the equivalence of models ( 29) and (30), from Theorem 16, we also obtained that model (29) has a positive order-1 periodic solution which is orbitally asymptotically stable.At the same time, it also implies that languages  and  are coexistent and have a stable equilibrium state under state-dependent impulsive control strategy.

Numerical Simulation and Discussion
. In this subsection, some numerical simulations are carried out to illustrate main results and the feasibility of state-dependent pulse feedback control measure.Firstly, we choose model parameters  =  = 0.5 and  = 0.7 and control parameters   = 0.8 and  = 0.15.From Figure 4, we see that state-dependent control measure plays an important role in preventing the disappearance of endangered language.Under statedependent impulsive control measure, the downward trend for language  was controlled effectively, and the fractions of language  and bilingual  are kept within reasonable levels.Furthermore, numerical simulations also show that the fractions of languages ,  and bilingual , though from different initial states, are stabilized in the same state.The corresponding numerical results are presented in Figures 4 and 5. Namely, model ( 29) has a positive order-1 periodic solution, which is orbitally asymptotically stable.This is certainly the case as shown in Theorems 13-16.Again from Figure 5(a)-5(c), the periodic solution is orbitally asymptotically stable instead of being Lyapunov asymptotically stable.In fact, it also shows exactly how different the two stabilities are.
Next, we investigate what effect has the choice of controlling parameters on the dynamical behavior of model ( 29) using numerical modeling method.We first choose   = 0.8 and parameter  to be 0.05, 0.15, 0.25, and 0.35, respectively.From Figure 6, we note that the length of times intervals between two control strategies are closely geared to the strength of control measure  and that the time interval increases with the increasing of .Again from Figures 6(b) and 6(c), it is obvious that the fractions of language  and bilingual  in population could maintain higher level for a long time due to larger .Of course, the cost of control measure is related to its strength.Furthermore, similar results can also be obtained from  = 0.15 and letting   be 0.9, 0.85, 0.8, and 0.7 in Figure 7.It is not hard to imagine, however, that the cost of control measure is very high if the fraction of monolingual speakers is incredibly low in the population.This is because if the fraction of monolingual speakers is high in the population, then it is extremely difficult to encourage monolingual speaker of the mainstream language to study endangered language.Of course, it is also not a good measure for the protection of endangered language.How do we choose appropriate parameters such that the fractions of language  and bilingual  are kept at reasonable levels with the minimal cost of control measure?It is an interesting problem; at the same time, it is extremely difficult.

Concluding Remarks
The dynamic behavior of two-language competitive model (1) with parameters (2) and  = 1 is analyzed systemically in this paper.By the linearization and Bendixson-Dulac theorem on dynamical system, some sufficient conditions on the globally asymptotical stability of the trivial equilibria, the existence, the local stability, and the global stability of positive equilibrium of model (1) are presented.The theoretical results show that languages  and  are coexistent by adjusting the values of model parameters  and .
And when considering the protection of endangered language, model (1) with ( 2) is extended to model (10) assessing the impact of continuous control measure.The theoretical results and numerical simulations indicate that the existence and stability of model (10) are sensitive to control parameter.Furthermore, using the optimal control theory, we derived and analyzed the conditions for optimality of the endangered language.Our results say that the optimal control has a very desirable effect for maintaining the fraction   29) with different state-dependent impulsive control parameters, where  = 0.15 and   = 0.9, 0.85, 0.8, and 0.75, respectively. of bilingual , and some comparisons between with and without control are made in the figures.
Finally, the dynamic behavior of model ( 1) with statedependent pulse control measure, that is, model (29), is studied in Section 4. The state-dependent pulse control measure causes the complexity for the dynamic behavior of model (29) such as frequent switching between states, irregular motion, and some uncertainties.This is the distinguished feature compared with continuous control measure.By the Poincaré map, analogue of Poincaré criterion, and qualitative analysis method, some sufficient conditions on the existence and orbitally asymptotical stability of positive order-1 periodic solution are presented.This amounts to the fact that we can control the fractions of languages ,  and bilingual  at reasonable levels by adjusting control parameters.Theoretical basis for finding a new measure to protect the endangered language is provided.

Figure 2 :
Figure2: The effect of control measure on the existence and stability of positive equilibrium for model(10) with  =  = 0.5,  = 0.7, and  = 1.
Similar to the discussion of Theorem 2, we can get the locally asymptotical stability of equilibrium   ( *  ,  *  ).Next, we discuss the global behavior of equilibrium   ( *  ,  * *  is unique.That is, model(11)has a unique positive equilibrium   ( *  ,  *  ) in the interior of Ω.