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

Authors: Lin-Fei Nie; Zhi-Dong Teng; Juan J. Nieto; Il Hyo Jung
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The dynamic behavior of a two-language competitive model is analyzed systemically in this paper. By the linearization and the Bendixson-Dulac theorem on dynamical system, some sufficient conditions on the globally asymptotical stability of the trivial equilibria and the existence and
the stability of the positive equilibrium of this model are presented. Nextly, in order to protect the endangered language, an optimal control problem relative to this model is explored. We derive some necessary conditions to solve the optimal control problem and present some numerical simulations using a Runge-Kutta fourth-order method. Finally, the languages competitive model is extended to this model assessing the impact of state-dependent pulse control strategy. Using the Poincaré map, differential inequality, and method of qualitative analysis, we prove the existence and stability of positive order-1 periodic solution for this control model. Numerical simulations are carried out to illustrate the main results and the feasibility of state-dependent impulsive control strategy.


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## Body

## 1. 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 [2–11] 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:
(1)dxdt=yPYX+bPBX-x(PXY+PXB),dydt=xPXY+bPBY-y(PYX+PYB),dbdt=xPXB+yPYB-b(PBY+PBX),
where capital lettersXandYdenote the two languages spoken in population; the uppercased letterBdenotes the group of bilingual speakers; and the lowercased lettersx,y, andb(withx+y+b=1) refer to the fraction of speakers of each of the languages in population and the fraction of bilingual speakers, respectively; parameterPXYdenotes the probability of a monolingual speaker of languageXbeing replaced in the population by a monolingual speaker of languageY, with analogous notation for the other parametersPYX,PXB,PBX,PYB, andPBY. 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 bysthe relative status of languageXand by1-sthat of languageY,
(2)PXB=ck(1-s)(1-x)α,PBX=PYX=c(1-k)s(1-y)α,PYB=cks(1-y)α,PBY=PXY=c(1-k)(1-s)(1-x)α,
wherecis a normalization factor related to the time scale,αis the power parameter, andkis the probability that the disappearance of a monolingual speaker of languageX(resp.,Y) will be compensated for by the appearance of a bilingual rather than by a monolingual speaker of languageY(resp.,X). 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 [17–23] 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.


## 2. Qualitative Analysis for Model (1)

Inserting the parameters formula (2) into model (1) and taking into accountx+y+b=1, we get the following reduced model (here, we assume thatα=1throughout the rest of this paper):
(3)dxdt=cs(1-k)(1-x)(1-y)-c(1-s)x(1-x)∶=f1(x,y),dydt=c(1-k)(1-s)(1-x)(1-y)-csy(1-y)∶=f2(x,y).By the biological background of model (3), we only consider model (3) in the biological meaning regionΩ={(x,y):x≥0,y≥0,x+y≤1}.The following theorem is on the nonnegativity of solution of model (3).Theorem 1.
The regionΩis positively invariant for model (3).The proof of Theorem1 is simple; we, therefore, omit it here.Now, we discussed the existence and stability of equilibria for model (3). The isocline corresponding todx/dt=0is the lines(1-k)(1-y)=(1-s)x. The isocline corresponding tody/dt=0is the line(1-k)(1-s)(1-x)=sy. Note that two isoclines do not have positive intersection point fors<(1-s)(1-k)or(1-s)<s(1-k)and have only one positive intersection point for1-k<(1-s)/s<1/(1-k). Namely, model (3) has only two trivial equilibria(1,0)and(0,1)fors<(1-s)(1-k)or(1-s)<s(1-k)and has two trivial equilibria(1,0);(0,1)and a positive equilibriumE(x*,y*)for1-k<(1-s)/s<1/(1-k), where
(4)x*=(1-k)s-(1-s)(1-k)21-s-(1-s)(1-k)2,y*=(1-k)(1-s)-(1-k)2ss(1-(1-k)2).The locally asymptotical stabilities of equilibria are determined by the eigenvalues of Jacobian matrixes of the linearization of model (3) around equilibria. It is easy to calculate that
(5)∂f1(x,y)∂x=-cs(1-k)(1-y)-c(1-s)(1-x)+c(1-s)x,∂f1(x,y)∂y=-cs(1-k)(1-x),∂f2(x,y)∂x=-c(1-k)(1-s)(1-y),∂f2(x,y)∂y=-c(1-k)(1-s)(1-x)-cs(1-y)+csy.Computing Jacobian matrixes from model (3) around(1,0),(0,1), and(x*,y*), we have that
(6)J(1,0)=(c(1-s)-cs(1-k)0-c(1-k)(1-s)-cs),J(0,1)=(-c(1-s)-cs(1-k)0cs-c(1-k)(1-s)),(7)J(x*,y*)=(-c(1-s)(1-x*)-cs(1-k)(1-x*)-c(1-k)(1-s)(1-y*)-cs(1-y*)).From (6), since(1-s)<s(1-k)implies that(1-s)(1-k)<s, it follows that the two eigenvalues ofJ(1,0)are negative and that one eigenvalue is positive and another one is negative forJ(0,1)when(1-s)<s(1-k).   Therefore, model (3) has a locally asymptotically stable node(1,0)and a saddle point(0,1)for(1-s)<s(1-k). Furthermore, since there is not equilibrium in interior ofΩfor(1-s)<s(1-k), model (3) has no periodic orbit lying entirely in the interior ofΩ. Therefore,(1,0)is globally asymptotically stable in the interior ofΩfor this case. Similarly, sinces<(1-s)(1-k)implies thats(1-k)<(1-s), we also have that(0,1)is globally asymptotically stable in the interior ofΩand(1,0)is a saddle point for model (3) whens<(1-s)(1-k).Consider the eigenvalues ofJ(x*,y*); letT=TrJ(x*,y*)andD=DetJ(x*,y*)be the trace and determinant ofJ(x*,y*), respectively. The eigenvalues ofJ(x*,y*)are provided by the following characteristic equation:
(8)λ2-Tλ+D=0.Note that when(1-k)<(1-s)/s<1/(1-k),
(9)T=-c(1-s)(1-x*)-cs(1-y*)<0,D=c2s(1-s)(1-x*)(1-y*)-c2s(1-s)(1-k)2(1-x*)(1-y*)>0.Hence, both eigenvalues have negative real part, and, hence,(x*,y*)is locally asymptotically stable when(1-k)<(1-s)/s<1/(1-k). Furthermore, when we mention that model (3) has no periodic orbits, refer to Lemma3.2of [16].To summarize the above discussion, we give some sufficient conditions for the existence and asymptotical stability of equilibria for model (3).Theorem 2.
For anyk,s∈(0,1), one of the following statements is valid.(a)
If(1-s)<s(1-k), then model (3) has only two trivial equilibria(1,0)and(0,1), where(0,1)is a saddle point and(1,0)is globally asymptotically stable in the interior ofΩ.(b)
Ifs<(1-s)(1-k), then model (3) has only two trivial equilibria(1,0)and(0,1), where(1,0)is a saddle point and(0,1)is globally asymptotically stable in the interior ofΩ.(c)
If(1-k)<(1-s)/s<1/(1-k), then model (3) has two saddle points(1,0),(0,1)and a globally asymptotically stable positive equilibrium(x*,y*).Remark 3.
From Theorem2 and the equivalence of models (1) and (3), we obtain that model (1) has a globally asymptotically stable node(1,0,0)and a saddle point(0,1,0)for(1-s)<s(1-k)and has a saddle point(1,0,0)and a globally asymptotically stable node(0,1,0)fors<(1-s)(1-k). Furthermore, if(1-k)<(1-s)/s<1/(1-k), model (1) has a coexistent equilibrium(x*,y*,1-x*-y*)which is globally asymptotically stable.Finally, we fix all parameters includingc,k,s, andαand carry out numerical investigations to confirm our main results obtained in this section. Firstly, we choosek=0.5,s=0.7,c=0.5, andα=1; it is easy to calculate that1-s=0.3<s(1-k)=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 languageXis permanent and languageYwill fade away in this case. Similar conclusion can be obtained from Figure 1(b) with parametersk=0.3,s=0.4,c=0.5, andα=1. However, if we choosek=s=c=0.5andα=1, then languagesXandYare coexistent and tend to positive equilibrium as shown in Figure 1(c).The asymptotical stability of equilibria of model (1) withc=0.5andα=1: (a) the stability of trivial equilibrium(1,0,0)withk=0.5ands=0.7; (b) the stability of trivial equilibrium(0,1,0)withk=0.3ands=0.7; and (c) the stability of positive equilibrium(x*,y*,1-x*-y*)withk=0.5ands=0.5.
(a)
(b)
(c)


## 3. 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 Theorem2 in Section 2, we know that(i)languageXis permanent and languageYand bilingual speakersBare extinct for(1-s)<s(1-k)and that(ii)languageYis permanent and languageXand bilingual speakersBare extinct fors<(1-s)(1-k). Considering the similarities of the two cases, we only need to consider case(i); that is,(1-s)<s(1-k).


### 3.1. Protecting of LanguageY  with Continuous Control for  (1-s)<s(1-k)

We consider model (1) with parameters (2) and continuous control measure. The control system is modeled by the following differential equations:
(10)dxdt=cs(1-k)(y+b)(1-y)-c(1-s)x(1-x)-cμx,dydt=c(1-s)(1-k)(x+b)(1-x)-csy(1-y),dbdt=ck(1-s)x(1-x)+cksy(1-y)-c(1-k)b[(1-s)(1-x)+s(1-y)]+cμx,
whereμis a controlled variable, which means that the fraction of languageXbecomes the bilingualBper unit of time. Note thatx+y+b=1; model (10) can be written as follows:
(11)dxdt=cs(1-k)(1-x)(1-y)-c(1-s)x(1-x)-cμx,dydt=c(1-s)(1-k)(1-y)(1-x)-csy(1-y).We denote the right-side of model (11) byf(x,y)andg(x,y), respectively. We, here, discuss the existence and asymptotical stability of positive equilibrium of model (11).Fromg(x,y)=0, it is easily shown that
(12)yμ=y(x)=(1-k)(1-s)(1-x)s.Substituting it inf(x,y), it follows thatf(1,y(1))=-μ<0and
(13)f(0,y(0))=cs(1-k)(1-(1-k)(1-s)s)>0
due to conditions(1-s)<s(1-k)andk,s∈(0,1). Hence, there is at least onexμ*∈(0,1)such thatf(xμ*,y(xμ*))=0. Furthermore, it is also easy to calculate that
(14)df(x,y(x))dx=c(1-k)[-s+2(1-k)(1-s)(1-x)]+c(1-s)(-1+2x)-cμ,(15)d2f(x,y(x))dx2=2(1-s)[1-(1-k)2]>0
for allx∈[0,1]. This together with (14) gives that
(16)df(x,y(x))dx<df(1,y(1))dx=(1-s)-s(1-k)-μ<0
for allx∈(0,1), where inequality(1-s)<s(1-k)is used. Hence,xμ*is unique. That is, model (11) has a unique positive equilibriumEμ(xμ*,yμ*)in the interior ofΩ.Similar to the discussion of Theorem2, we can get the locally asymptotical stability of equilibriumEμ(xμ*,yμ*). Next, we discuss the global behavior of equilibriumEμ(xμ*,yμ*). LetB(x,y)=(1-x)-1(1-y)-1; we have
(17)∂(Bf)∂x+∂(Bg)∂y=-c(1-s)1-y-cs1-x-cμ(1-x)2(1-y)<0.
So model (11) has no closed orbit lying entirely in the interior ofΩ. We, therefore, can show from the above discussion that equilibriumEμ(xμ*,yμ*)is globally asymptotically stable in the interior ofΩ. The result can be written by the following theorem.Theorem 4.
For anyμ∈(0,1), if(1-s)<s(1-k), then model (11) has a globally asymptotically stable positive equilibriumEμ(xμ*,yμ*)in the interior ofΩ.Remark 5.
From Theorem2, it follows that languageXis permanent and that languageYand bilingual speakersBwill eventually disappear for(1-s)<s(1-k)in model (1). However, if we introduce a control variableμ(no matter how small it is) in model (1), languagesXandYare coexistent and tend to positive equilibrium(xμ*,yμ*,1-xμ*-yμ*). The coexistent state, of course, depends upon the controlled strengthμ. This implies thatμis a sensitive controlled parameter for the protection of endangered language.


### 3.2. 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 variableμ(t)∈Uad; here,Uad={μ:μ(t)ismeasurable,0≤μ(t)≤0.9forallt∈[0,tfinal]}indicates an admissible control. In this optimal problem, we assume a restriction on the control variableμ(t)such that0≤μ(t)≤0.9, because conversion of all of languageXat one time is impossible. In case of no control, the fraction of languageXincreases while the fractions of languageYand bilingualBdie out. Therefore, the biological meaning of an optimal control in this problem is that the adequate levels for the fractions of languageXand bilingualBare built.Now, we consider an optimal control problem to maximize the objective functional(18)J(μ)=∫0tfinal‍(C1b-12C2μ2)dt
subject to model (10). The first term represents the benefit of bilingualB, and the other term is systemic cost of control measure. The positive constantsC1andC2balance the size of the termsBandμ. Our goal is maximizing the fraction of bilingualBand minimizing the systemic cost to the control measure. This seeks an optimal controlμ*such that
(19)J(μ*)=max{J(μ):μ∈Uad}
subject to model (10). It is obvious that the integrand of objective functionalJis a convex function of control variableμand 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 HamiltonianHwith respect toμas follows:
(20)H=C1b-12C2μ2+λ1[cs(1-k)(y+b)(1-y)-c(1-s)x(1-x)-cμx]+λ2[c(1-s)(1-k)(x+b)(1-x)-csy(1-y)]+λ3{ck(1-s)x(1-x)+cksy(1-y)-c(1-k)b×[(1-s)(1-x)+s(1-y)]+cμxck(1-s)},
whereλ1,λ2, andλ3are adjoint variables.In the following theorem, we derive the necessary conditions for the optimal control problem.Theorem 6.
Let(x*,y*,b*)be an optimal state solution with associated optimal control variableμ*for the maximized object functionalJ(μ)subject to control model (10). Then, for model (10), there exist adjoint variablesλ1,λ2, andλ3such that
(21)dλ1dt=λ1[c(1-s)(1-x*)-c(1-s)x*+cμ*]+λ2[c(1-s)(1-k)(x*+b*)-c(1-s)×(1-k)(1-x*)]+λ3[ck(1-s)x*-ck(1-s)(1-x*)-c(1-k)(1-s)b*-cμ*],(22)dλ2dt=λ1[cs(1-k)(y*+b*)-cs(1-k)(1-y*)]+λ2[cs(1-y*)-csy*]+λ3[cksy*-cks(1-y*)-cs(1-k)b*],(23)dλ3dt=-C1-λ1cs(1-k)(1-y*)-λ2c(1-s)×(1-k)(1-x*)+λ3c(1-k)[(1-s)(1-x*)+s(1-y*)]
with transversality conditionsλi(tfinal)=0,i=1,2,3. Furthermore, the optimal control is given by
(24)μ*(t)=min{0.9,max{0,(λ3-λ1)cx*A}}.Proof.
From Pontryagin’s Maximum Principle, adjoint variablesλi(i=1,2,3)can be written as
(25)dλ1dt=-∂H∂x,dλ2dt=-∂H∂y,dλ3dt=-∂H∂b.
These are just differential equations (21)–(23) with transversality conditionsλi(tfinal)=0(i=1,2,3). Furthermore, by optimality condition, we have that
(26)∂H∂μ|μ=μ*=-C2μ*-cλ1x*+cλ3x*=0.
This shows that(27)μ*=(λ3-λ1)cx*C2.
Using the property of control space, we get that(28)μ*={0,(λ3-λ1)cx*C2≤0,(λ3-λ1)cx*C2,0<(λ3-λ1)cx*C2<0.9,0.9,(λ3-λ1)cx*C2≥0.9.
This can be rewritten in compact notation, which is just (24). This completes the proof.


### 3.3. 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,c=k=0.5ands=0.7. Besides that, we choose control variableμ=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(xμ*,yμ*,bμ*)=(0.4194,0.1244,0.4562). Furthermore, from the discussion of Theorem 4, it follows that valuesxμ*andyμ*will decrease and increase with increase in control strengthμ, respectively, which is shown in Figures 2(b) and 2(c). The strong consistency between theoretical result and real situation is obviously observed.The effect of control measure on the existence and stability of positive equilibrium for model (10) withc=k=0.5,s=0.7, andα=1.
(a)
(b)
(c)In addition, in Figure2(c), it is interesting to note that control strengthμis close to 1 (whereμ=0.95), but languageXruns around 0.2. This means that, no matter how strong the control strengthμis, languageXwill not fade. Namely, control measure can only protect endangered languageY, but not result in extinction of languageX. Actually, in the real world, the own characteristics are very important factors in determining the development of languages.The plots in Figure3(a) show three adjoint variablesλ1,λ2, andλ3in 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 languagesX,Yand bilingualBin model (10) without and with continuous control, respectively. We see that the fractions of languageYand bilingualBin population decrease more when there is no control. In this case, most  of this population goes to languageX. If we apply the continuous control measure, however, the fraction of languageYslowly falls, the fraction of languageXdecreases quite a lot, and the fraction of bilingualBis quite greater than the fraction in the case without control, since our main object is maximizing the fraction of bilingualB. In Figure 3(c), the controlμis plotted as a function of time for three different values of weight factorC2:0.045,0.1, and0.5. The control variableμfor the associated weight factorμ=0.045is much larger than the other two values. Note that, in general, asC2decreases, the amount ofμincreases. The same results can also be obtained from the expression ofμin (27). The associated weight factorC2also plays a significant role in keeping the balance of the size of fraction an optimal problem.The optimal adjoint variables, states, and control variable for the optimal control problem withc=k=0.5,s=0.7,α=1, andC1=0.1: (a) the optimal adjoint variablesλ1,λ2, andλ3; (b) the optimal states of languagesX,Yand bilingualB; and (c) the control variableμwith different weight factorC2.
(a)
(b)
(c)


## 3.1. Protecting of LanguageY  with Continuous Control for  (1-s)<s(1-k)

We consider model (1) with parameters (2) and continuous control measure. The control system is modeled by the following differential equations:
(10)dxdt=cs(1-k)(y+b)(1-y)-c(1-s)x(1-x)-cμx,dydt=c(1-s)(1-k)(x+b)(1-x)-csy(1-y),dbdt=ck(1-s)x(1-x)+cksy(1-y)-c(1-k)b[(1-s)(1-x)+s(1-y)]+cμx,
whereμis a controlled variable, which means that the fraction of languageXbecomes the bilingualBper unit of time. Note thatx+y+b=1; model (10) can be written as follows:
(11)dxdt=cs(1-k)(1-x)(1-y)-c(1-s)x(1-x)-cμx,dydt=c(1-s)(1-k)(1-y)(1-x)-csy(1-y).We denote the right-side of model (11) byf(x,y)andg(x,y), respectively. We, here, discuss the existence and asymptotical stability of positive equilibrium of model (11).Fromg(x,y)=0, it is easily shown that
(12)yμ=y(x)=(1-k)(1-s)(1-x)s.Substituting it inf(x,y), it follows thatf(1,y(1))=-μ<0and
(13)f(0,y(0))=cs(1-k)(1-(1-k)(1-s)s)>0
due to conditions(1-s)<s(1-k)andk,s∈(0,1). Hence, there is at least onexμ*∈(0,1)such thatf(xμ*,y(xμ*))=0. Furthermore, it is also easy to calculate that
(14)df(x,y(x))dx=c(1-k)[-s+2(1-k)(1-s)(1-x)]+c(1-s)(-1+2x)-cμ,(15)d2f(x,y(x))dx2=2(1-s)[1-(1-k)2]>0
for allx∈[0,1]. This together with (14) gives that
(16)df(x,y(x))dx<df(1,y(1))dx=(1-s)-s(1-k)-μ<0
for allx∈(0,1), where inequality(1-s)<s(1-k)is used. Hence,xμ*is unique. That is, model (11) has a unique positive equilibriumEμ(xμ*,yμ*)in the interior ofΩ.Similar to the discussion of Theorem2, we can get the locally asymptotical stability of equilibriumEμ(xμ*,yμ*). Next, we discuss the global behavior of equilibriumEμ(xμ*,yμ*). LetB(x,y)=(1-x)-1(1-y)-1; we have
(17)∂(Bf)∂x+∂(Bg)∂y=-c(1-s)1-y-cs1-x-cμ(1-x)2(1-y)<0.
So model (11) has no closed orbit lying entirely in the interior ofΩ. We, therefore, can show from the above discussion that equilibriumEμ(xμ*,yμ*)is globally asymptotically stable in the interior ofΩ. The result can be written by the following theorem.Theorem 4.
For anyμ∈(0,1), if(1-s)<s(1-k), then model (11) has a globally asymptotically stable positive equilibriumEμ(xμ*,yμ*)in the interior ofΩ.Remark 5.
From Theorem2, it follows that languageXis permanent and that languageYand bilingual speakersBwill eventually disappear for(1-s)<s(1-k)in model (1). However, if we introduce a control variableμ(no matter how small it is) in model (1), languagesXandYare coexistent and tend to positive equilibrium(xμ*,yμ*,1-xμ*-yμ*). The coexistent state, of course, depends upon the controlled strengthμ. This implies thatμis a sensitive controlled parameter for the protection of endangered language.


## 3.2. 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 variableμ(t)∈Uad; here,Uad={μ:μ(t)ismeasurable,0≤μ(t)≤0.9forallt∈[0,tfinal]}indicates an admissible control. In this optimal problem, we assume a restriction on the control variableμ(t)such that0≤μ(t)≤0.9, because conversion of all of languageXat one time is impossible. In case of no control, the fraction of languageXincreases while the fractions of languageYand bilingualBdie out. Therefore, the biological meaning of an optimal control in this problem is that the adequate levels for the fractions of languageXand bilingualBare built.Now, we consider an optimal control problem to maximize the objective functional(18)J(μ)=∫0tfinal‍(C1b-12C2μ2)dt
subject to model (10). The first term represents the benefit of bilingualB, and the other term is systemic cost of control measure. The positive constantsC1andC2balance the size of the termsBandμ. Our goal is maximizing the fraction of bilingualBand minimizing the systemic cost to the control measure. This seeks an optimal controlμ*such that
(19)J(μ*)=max{J(μ):μ∈Uad}
subject to model (10). It is obvious that the integrand of objective functionalJis a convex function of control variableμand 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 HamiltonianHwith respect toμas follows:
(20)H=C1b-12C2μ2+λ1[cs(1-k)(y+b)(1-y)-c(1-s)x(1-x)-cμx]+λ2[c(1-s)(1-k)(x+b)(1-x)-csy(1-y)]+λ3{ck(1-s)x(1-x)+cksy(1-y)-c(1-k)b×[(1-s)(1-x)+s(1-y)]+cμxck(1-s)},
whereλ1,λ2, andλ3are adjoint variables.In the following theorem, we derive the necessary conditions for the optimal control problem.Theorem 6.
Let(x*,y*,b*)be an optimal state solution with associated optimal control variableμ*for the maximized object functionalJ(μ)subject to control model (10). Then, for model (10), there exist adjoint variablesλ1,λ2, andλ3such that
(21)dλ1dt=λ1[c(1-s)(1-x*)-c(1-s)x*+cμ*]+λ2[c(1-s)(1-k)(x*+b*)-c(1-s)×(1-k)(1-x*)]+λ3[ck(1-s)x*-ck(1-s)(1-x*)-c(1-k)(1-s)b*-cμ*],(22)dλ2dt=λ1[cs(1-k)(y*+b*)-cs(1-k)(1-y*)]+λ2[cs(1-y*)-csy*]+λ3[cksy*-cks(1-y*)-cs(1-k)b*],(23)dλ3dt=-C1-λ1cs(1-k)(1-y*)-λ2c(1-s)×(1-k)(1-x*)+λ3c(1-k)[(1-s)(1-x*)+s(1-y*)]
with transversality conditionsλi(tfinal)=0,i=1,2,3. Furthermore, the optimal control is given by
(24)μ*(t)=min{0.9,max{0,(λ3-λ1)cx*A}}.Proof.
From Pontryagin’s Maximum Principle, adjoint variablesλi(i=1,2,3)can be written as
(25)dλ1dt=-∂H∂x,dλ2dt=-∂H∂y,dλ3dt=-∂H∂b.
These are just differential equations (21)–(23) with transversality conditionsλi(tfinal)=0(i=1,2,3). Furthermore, by optimality condition, we have that
(26)∂H∂μ|μ=μ*=-C2μ*-cλ1x*+cλ3x*=0.
This shows that(27)μ*=(λ3-λ1)cx*C2.
Using the property of control space, we get that(28)μ*={0,(λ3-λ1)cx*C2≤0,(λ3-λ1)cx*C2,0<(λ3-λ1)cx*C2<0.9,0.9,(λ3-λ1)cx*C2≥0.9.
This can be rewritten in compact notation, which is just (24). This completes the proof.


## 3.3. 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,c=k=0.5ands=0.7. Besides that, we choose control variableμ=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(xμ*,yμ*,bμ*)=(0.4194,0.1244,0.4562). Furthermore, from the discussion of Theorem 4, it follows that valuesxμ*andyμ*will decrease and increase with increase in control strengthμ, respectively, which is shown in Figures 2(b) and 2(c). The strong consistency between theoretical result and real situation is obviously observed.The effect of control measure on the existence and stability of positive equilibrium for model (10) withc=k=0.5,s=0.7, andα=1.
(a)
(b)
(c)In addition, in Figure2(c), it is interesting to note that control strengthμis close to 1 (whereμ=0.95), but languageXruns around 0.2. This means that, no matter how strong the control strengthμis, languageXwill not fade. Namely, control measure can only protect endangered languageY, but not result in extinction of languageX. Actually, in the real world, the own characteristics are very important factors in determining the development of languages.The plots in Figure3(a) show three adjoint variablesλ1,λ2, andλ3in 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 languagesX,Yand bilingualBin model (10) without and with continuous control, respectively. We see that the fractions of languageYand bilingualBin population decrease more when there is no control. In this case, most  of this population goes to languageX. If we apply the continuous control measure, however, the fraction of languageYslowly falls, the fraction of languageXdecreases quite a lot, and the fraction of bilingualBis quite greater than the fraction in the case without control, since our main object is maximizing the fraction of bilingualB. In Figure 3(c), the controlμis plotted as a function of time for three different values of weight factorC2:0.045,0.1, and0.5. The control variableμfor the associated weight factorμ=0.045is much larger than the other two values. Note that, in general, asC2decreases, the amount ofμincreases. The same results can also be obtained from the expression ofμin (27). The associated weight factorC2also plays a significant role in keeping the balance of the size of fraction an optimal problem.The optimal adjoint variables, states, and control variable for the optimal control problem withc=k=0.5,s=0.7,α=1, andC1=0.1: (a) the optimal adjoint variablesλ1,λ2, andλ3; (b) the optimal states of languagesX,Yand bilingualB; and (c) the control variableμwith different weight factorC2.
(a)
(b)
(c)


## 4. Protection of LanguageY  with Impulsive Control for  (1-s)<s(1-k)

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:(29)dxdt=yPYX+bPBX-x(PXY+PXB),dydt=xPXY+bPBY-y(PYX+PYB),dbdt=xPXB+yPYB-b(PBY+PBX),x<EX,x(t+)=(1-θ)x(t),y(t+)=y(t),b(t+)=b(t)+θx(t),x=EX.The meaning of model (29) is as follows: when the fraction of languageXreaches the critical threshold valueEXat timetE, controlling measure (for example, encouraging some speaker of languageXto study languageY) is taken and the fractions of languageXand bilingual speakersBimmediately become(1-θ)x(tE)andb(tE)+θx(tE), respectively.Remark 7.
It is obvious that the fractions of languageYand bilingualBare rather small and in danger of becoming extinct when the fraction of languageXreaches the critical threshold valueEX. In this case, the effective measure is taken to prevent the loss of languageY. The times of control measures are obviously related to the state of languageX.Remark 8.
The critical threshold valueEXrepresents the fraction of monolingual speakersXin population, andθrepresents the strength of control measure. Numerical simulations in Section 4.3 show that these are crucial parameters in model (29).For model (29), which is equivalent to the following reduced model since the population has a constant sizex+y+b=1,
(30)dxdt=c(1-x)[(1-k)s(x+b)-x(1-s)],dbdt=c(1-x)[kx-b(1-s-k)]-bcs(x+b),x≠EX,x(t+)=(1-θ)x(t),b(t+)=b(t)+θx(t),x=EX.By the biological background of model (30), we only consider model in the biological meaning regionΩ0={(x,b):x≥0,b≥0,x+b≤1}. Obviously, the global existence and uniqueness of solution of model (30) are guaranteed by the smoothness properties of the right-side of model (30); for more details, see Lakshmikantham et al. [29].


### 4.1. Preliminaries

To discuss the dynamic behavior of model (30), we define two cross-sections to the phase space of model (30) by sections
(31)Γθ={(x,b):x=(1-θ)EX,b≥0},ΓE={(x,b):x=EX,b≥0}.Suppose that pointAn(EX,bn)is on sectionΓE; then, trajectory
(32)O+(An,tn)={(x(t),b(t)):(x(t),b(t))∈Ω0,t≥tn,x(tn)=EX,b(tn)=bn}
jumps to pointAn+((1-θ)EX,b^n)on sectionΓθdue to pulse effectsx(tn+)=(1-m)x(tn)andb(tn+)=b(tn)+θx(tn), then intersects sectionΓEat pointAn+1(EX,bn+1), and finally jumps to pointAn+1+((1-θ)EX,b^n+1)on sectionΓθagain, wherebn+1is decided bybnand parametersθ,EX. Therefore, we defined a Poincaré map of sectionΓEas follows:
(33)bn+1=F(bn,θ,EX).Letz(t)=(x(t),b(t))be a solution of model (30) with initial conditionz0=z(t0)=((1-θ)EX,b^0). TrajectoryO+(z0,t0)starts from pointA0((1-θ)EX,b^0); first, it intersects sectionΓEat pointB1(EX,b1), then jumps to pointA1((1-θ)EX,b^1)on sectionΓθdue to pulse effects, and finally reaches pointB2(Ex,b2)on sectionΓEagain. Repeating the above process, we have two points’ sequences{An-1((1-θ)EX,b^n-1)}and{Bn(EX,bn)}(n=1,2,…). We notice that the coordinates satisfy the relationb^n=bn+θEX(n=1,2,…).Let𝕊be an arbitrary set inℝ2, and letPbe an arbitrary point inℝ2. The distance between pointPand set𝕊is defined byd(P,𝕊)=infP0∈𝕊|P-P0|. For the convenience of statement in the rest of this paper, we introduce some definitions.Definition 9.
TrajectoryO+(z0,t0)of model (30) is said to be order-kperiodic if there exists a positive integerk≥1such thatkis the smallest integer forb^0=b^k.Definition 10 (orbital stability [30]).
TrajectoryO+(X0,t0)is said to be orbitally stable if, given thatε>0, there exists a constantδ=δ(ε)>0such that, for other solutionz*(t)of model (30), whend(z*(t0),O+(z0,t0))<δ, one has thatd(z*(t),O+(z0,t0))<εfor allt>t0.Definition 11 (orbitally asymptotical stability [30]).
TrajectoryO+(z0,t0)is said to be orbitally asymptotically stable if it is orbitally stable, and there exists a constantη>0such that, for any other solutionz*(t)of model (30), whend(z*(t0),O+(z0,t0))<η,limt→∞d(z*(t),O+(z0,t0))=0.Next, we consider the following autonomous model with pulse effects:(34)dxdt=f(x,y),dydt=g(x,y),φ(x,y)≠0,Δx=ξ(x,y),Δy=η(x,y),φ(x,y)=0,
wheref(x,y)andg(x,y)are continuous differential functions defined onℝ2andφ(x,y)is a sufficiently smooth function with∇φ(x,y)≠0. Let(μ(t),ν(t))be a positiveT-periodic solution of model (34). The following result comes from Corollary2of Theorem1in [31].Lemma 12 (analogue of Poincaré criterion).
If the Floquet multiplierμsatisfies condition|μ|<1, where
(35)μ=∏j=1n‍κjexp{∫0T‍[∂f(μ(t),ν(t))∂x+∂g(μ(t),ν(t))∂y]dt}
with
(36)κj=((∂η∂y∂φ∂x-∂η∂x∂φ∂y+∂φ∂x)f++(∂ξ∂x∂φ∂y-∂ξ∂y∂φ∂x+∂φ∂y)g+)×(∂φ∂xf+∂φ∂yg)-1
andf,g,∂ξ/∂x,∂ξ/∂y,∂η/∂x,∂η/∂y,∂φ/∂x, and∂φ/∂yhave been calculated at the point(μ(τj),ν(τj)),f+=f(μ(τj+),ν(τj+)),g+=g(μ(τj+),ν(τj+)), andτj(j∈N)is the time of thejth jump, then,(μ(t),ν(t))is orbitally asymptotically stable.


### 4.2. Main Results

On the existence of positive order-1 periodic solution for model (30), we have the following theorem.Theorem 13.
For anyθ∈(0,1)andEX∈(0,1), model (30) admits a positive order-1 periodic solution.Proof.
Let pointA0((1-θ)EX,s^0)∈Γθfor sufficiently smalls^0withs^0≤θEX. In view of the geometrical structure of the phase space of model (30), trajectoryO+(A0,t0)of model (30) starts from initial pointA0and intersects sectionΓEat pointB1(EX,s1). And then, trajectoryO+(A0,t0)jumps to pointA1(s^1,(1-θ)EX)on sectionΓθdue to pulse control effectsx(t+)=(1-θ)x(t)andb(t+)=b(t)+θx(t). Sinces^0≤θEX, it follows that pointA1is above pointA0. Furthermore, pointB2is above pointB1. Otherwise, this is a contradiction with the uniqueness of solution for model (30). By (33), we have thats2=F(s1,θ,EX)and
(37)F(s1,θ,EX)-s1=s2-s1>0.
On the other hand, letλ0=1-(1-θ)EX. For critical pointC0((1-θ)EX,λ0), trajectoryO+(C0,t0)starts from initial pointC0; it intersects sectionΓEat pointD1(EX,λ1), then jumps to pointC1((1-m)EX,λ1+)on sectionΓθ, and finally reaches pointD2(EX,λ2)on sectionΓEagain. Sincex+b≤1, one has that(1-m)EX+λ1+≤1=(1-θ)EX+λ0. That is,λ1+≤λ0. Ifλ1+=λ0, namely, pointsC0andC1coincide, then model (30) has a positive order-1 periodic solution. Otherwise,λ1+<λ0; that is, pointC0is above pointC1, and pointD1is above pointD2. By (33), we can get thatλ2=F(λ1,θ,EX)and
(38)F(λ1,θ,EX)-λ1=λ2-λ1<0.
This together with (37) yields that Poincaré map (33) has a fixed point; that is, model (30) has a positive order-1 periodic solution. This completes the proof.Now, on the orbital stability of positive order-1 periodic solution of model (30), we have the following result.Theorem 14.
Let(φ(t),ψ(t))be a positive order-1 periodic solution of model (30) with periodT. For anyθ∈(0,1)andEX∈(0,1), if the Floquet multiplierμsatisfies condition|μ|<1, where
(39)|μ|=|κ|exp{∫0T‍[c(1-φ(t))(3s+k-ks-2)-cs(φ(t)+ψ(t))-csψ(t)]dt∫0T‍[c(1-φ(t))(3s+k-ks-2)}
with
(40)κ=(1-k)s(EX+ψ(T))-(1-θ)EX(1-s)(1-k)s(EX+ψ(T))-EX(1-s),
then(φ(t),ψ(t))is orbitally asymptotically stable.Proof.
Suppose that(φ(t),ψ(t))intersects sectionsΓθandΓEat pointsC+((1-θ)EX,θEX+ψ(T))andC(EX,ψ(T)), respectively. Comparing with model (34), we have that
(41)f(x,b)=c(1-x)[(1-k)s(x+b)-x(1-s)],g(x,b)=c(1-x)[kx-b(1-s-k)]-bcs(x+b),
andξ(x,b)=-θx,η(x,b)=θx,ϕ(S,I)=x-EX,(φ(T),ψ(T))=(EX,ψ(T)), and(φ(T+),ψ(T+))=((1-θ)EX,ψ(T)+θEX). Thus,
(42)∂f∂x=-c[(1-k)s(x+b)-x(1-s)]+c(1-x)(2s-ks-1),∂g∂b=-c(1-x)(1-s-k)-cs(x+b)-csb,∂ξ∂x=-θ,∂η∂x=θ,∂ϕ∂x=1,∂ξ∂b=∂η∂b=∂ϕ∂b=0.
Furthermore, it follows from (42) that
(43)κ=((∂η∂b∂ϕ∂x-∂η∂x∂ϕ∂b+∂ϕ∂x)f++(∂ξ∂x∂ϕ∂b-∂ξ∂b∂ϕ∂x+∂ϕ∂b)g+)×(∂ϕ∂xf+∂ϕ∂bg)-1=f+(φ(T+),ψ(T+))f(φ(T),ψ(T))=((1-(1-θ)EX)[(1-k)s(EX+ψ(T))-(1-θ)EX(1-s)](1-(1-θ)EX)[(1-k)s(EX+ψ(T)))×((1-EX)[(1-k)s(EX+ψ(T))-EX(1-s)(1-k)s(EX](1-EX)[(1-k)s(EX)-1,(44)μ=κexp{∫0T[-c((1-k)s(φ(t)+ψ(t))‍-φ(t)(1-s))+c(1-φ(t))×(2s-ks-1)-c(1-φ(t))(1-s-k)-cs(φ(t)+ψ(t))-csψ(t)]dt∫0T[-c((1-k)s(φ(t)+ψ(t))‍}.
On the other hand, integrating both sides of the first equation of model (30) along the orbitC+C^, we have that
(45)ln1-EX1-(1-θ)EX=∫(1-θ)EXEX‍dx1-x=-∫0T‍c[(1-k)s(φ(t)+ψ(t))-φ(t)(1-s)]dt.
From (43)–(45), we can obtain that
(46)|μ|=|(1-k)s(EX+ψ(T))-(1-θ)EX(1-s)(1-k)s(EX+ψ(T))-EX(1-s)|×exp{∫0T‍[c(1-φ(t))(3s+k-ks-2)-cs(φ(t)+ψ(t))-csψ(t)]dt∫0T‍}.
By condition (39), we see that model (30) satisfies all conditions of Lemma 12. Therefore, order-1 periodic solution(φ(t),ψ(t))of model (30) is orbitally asymptotically stable and has asymptotic phase property. This completes the proof.Remark 15.
From Theorem14, 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).Theorem 16.
For anyθ∈(0,1)andEX∈(0,1), model (30) has a positive order-1 periodic solution which is orbitally asymptotically stable.Proof.
Letλ*=1-(1-θ)EX, and suppose that the trajectoryO+(C0,t0)of model (30) starts from critical pointC0((1-θ)EX,λ*)and intersects sectionΓEat pointC*(EX,β*). In view of the geometrical construction of phase space of model (30) and that(1,0)is a globally asymptotically stable node, we obtain that trajectory of model (30) which starts from point((1-θ)EX,b)withb∈(0,λ*)will intersect sectionΓEat point(EX,b^); then,b^∈(0,β*). So we only need to consider trajectories of model (30) which start from the point(EX,b)on sectionΓE, whereb∈(0,β*).
Suppose that trajectoryO+(D1,t0)of model (30) which starts from initial pointD1(EX,b1)(0<b1<β*) jumps to pointD1+((1-θ)EX,b^1)on sectionΓθdue to pulse effects and then reaches sectionΓEat pointD2(EX,b2), whereb^1∈(0,λ*)andb2∈(0,β*). Repeating the above process, we have pulse points sequences{Dn(EX,bn)}and{Dn+((1-θ)EX,b^n)}, wherebn∈(0,β*)andb^n∈(0,λ*). Furthermore, this follows from Poincaré map (33) thatbn+1=F(bn,θ,EX)(n=1,2,…). On the other hand, for any two pointsDi(EX,bi)andDj(EX,bj)on sectionΓE, wherebi,bj∈(0,β*)andbi<bj, in view of pulse effects, pointDi+((1-θ)EX,b^i)is below pointDj+((1-θ)EX,b^j). And then, trajectoriesO+(Di,t0)andO+(Dj,t0)intersect sectionΓEat pointsDi+1(EX,bi+1)andDj+1(EX,bj+1), respectively. Therefore, from the geometrical construction of the phase space of model (30), we claim that
(47)0<bi+1<bj+1<β*.
In fact, if inequality (47) does not hold, that is,bi+1≥bj+1, then, it follows that pointDi+1is above pointDj+1or that two points coincide. Furthermore, we can obtain that trajectoriesO+(Di,t0)andO+(Dj,t0)intersect at a pointD*(x^,b^). It is indicated that there are two different solutions which start from pointD*. This is a contradiction with the uniqueness of solution of model (30). So inequality (47) is thus valid.
Now, for anyb0∈(0,β*), from Poincaré map (33) of sectionΓE, we haveb1=F(b0,θ,EX),b2=F(b1,θ,EX), andbn+1=F(bn,θ,EX)(n=2,3,4,…). In particular, ifb0=b1, then model (30) has a positive order-1 periodic solution, ifb0≠b1, then it follows from the fact (47) thatb0≠b2.
Next, we discuss the general circumstance; that is,b0≠b1≠b2≠⋯≠bk(k>2).(a)
Ifb0<b1, from (47), we obtain thatb1<b2. Then,b0<b1<b2. In this case, repeating the above process, we have that
(48)0<b0<b1<b2<b3⋯<β*.(b)
Ifb0>b1, similar to (a), we have that
(49)0<⋯<b3<b2<b1<b0<β*.
Therefore, in case (a), we can get thatlimn→∞bn=η. This implies that model (30) has an orbitally asymptotically stable positive order-1 periodic solution. Similarly, we can obtain the same result in case (b). This completes the proof.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 languagesXandYare coexistent and have a stable equilibrium state under state-dependent impulsive control strategy.Remark 18.
Similar results can also be obtained from the case thats<(1-s)(1-k); we, hence, omit them here.


### 4.3. 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 parametersc=k=0.5ands=0.7and control parametersEX=0.8andθ=0.15. From Figure 4, we see that state-dependent control measure plays an important role in preventing the disappearance of endangered language. Under state-dependent impulsive control measure, the downward trend for languageYwas controlled effectively, and the fractions of languageXand bilingualBare kept within reasonable levels. Furthermore, numerical simulations also show that the fractions of languagesX,Yand bilingualB, 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.The contrast of the dynamical behaviors of model (29) with state-dependent impulsive control measure and without control, wherec=k=0.5,s=0.7,EX=0.8, andθ=0.15.
(a)
(b)
(c)The orbitally asymptotical stability of positive order-1 periodic solution for model (29).
(a)
(b)
(c)Next, we investigate what effect has the choice of controlling parameters on the dynamical behavior of model (29) using numerical modeling method. We first chooseEX=0.8and parameterθto be0.05,0.15,0.25, and0.35, respectively. From Figure 6, we note that the length of times intervals between two control strategies are closely geared to the strength of control measureθand that the time interval increases with the increasing ofθ. Again from Figures 6(b) and 6(c), it is obvious that the fractions of languageYand bilingualBin population could maintain higher level for a long time due to largerθ. Of course, the cost of control measure is related to its strength. Furthermore, similar results can also be obtained fromθ=0.15and lettingEXbe0.9,0.85,0.8, and0.7in 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 languageYand bilingualBare kept at reasonable levels with the minimal cost of control measure? It is an interesting problem; at the same time, it is extremely difficult.The contrast of the dynamical behaviors of model (29) with different state-dependent impulsive control parameters, whereEX=0.8andθ=0.05, 0.15, 0.25, and 0.35, respectively.
(a)
(b)
(c)The contrast of the dynamical behaviors of model (29) with different state-dependent impulsive control parameters, whereθ=0.15andEX=0.9, 0.85, 0.8, and 0.75, respectively.
(a)
(b)
(c)


## 4.1. Preliminaries

To discuss the dynamic behavior of model (30), we define two cross-sections to the phase space of model (30) by sections
(31)Γθ={(x,b):x=(1-θ)EX,b≥0},ΓE={(x,b):x=EX,b≥0}.Suppose that pointAn(EX,bn)is on sectionΓE; then, trajectory
(32)O+(An,tn)={(x(t),b(t)):(x(t),b(t))∈Ω0,t≥tn,x(tn)=EX,b(tn)=bn}
jumps to pointAn+((1-θ)EX,b^n)on sectionΓθdue to pulse effectsx(tn+)=(1-m)x(tn)andb(tn+)=b(tn)+θx(tn), then intersects sectionΓEat pointAn+1(EX,bn+1), and finally jumps to pointAn+1+((1-θ)EX,b^n+1)on sectionΓθagain, wherebn+1is decided bybnand parametersθ,EX. Therefore, we defined a Poincaré map of sectionΓEas follows:
(33)bn+1=F(bn,θ,EX).Letz(t)=(x(t),b(t))be a solution of model (30) with initial conditionz0=z(t0)=((1-θ)EX,b^0). TrajectoryO+(z0,t0)starts from pointA0((1-θ)EX,b^0); first, it intersects sectionΓEat pointB1(EX,b1), then jumps to pointA1((1-θ)EX,b^1)on sectionΓθdue to pulse effects, and finally reaches pointB2(Ex,b2)on sectionΓEagain. Repeating the above process, we have two points’ sequences{An-1((1-θ)EX,b^n-1)}and{Bn(EX,bn)}(n=1,2,…). We notice that the coordinates satisfy the relationb^n=bn+θEX(n=1,2,…).Let𝕊be an arbitrary set inℝ2, and letPbe an arbitrary point inℝ2. The distance between pointPand set𝕊is defined byd(P,𝕊)=infP0∈𝕊|P-P0|. For the convenience of statement in the rest of this paper, we introduce some definitions.Definition 9.
TrajectoryO+(z0,t0)of model (30) is said to be order-kperiodic if there exists a positive integerk≥1such thatkis the smallest integer forb^0=b^k.Definition 10 (orbital stability [30]).
TrajectoryO+(X0,t0)is said to be orbitally stable if, given thatε>0, there exists a constantδ=δ(ε)>0such that, for other solutionz*(t)of model (30), whend(z*(t0),O+(z0,t0))<δ, one has thatd(z*(t),O+(z0,t0))<εfor allt>t0.Definition 11 (orbitally asymptotical stability [30]).
TrajectoryO+(z0,t0)is said to be orbitally asymptotically stable if it is orbitally stable, and there exists a constantη>0such that, for any other solutionz*(t)of model (30), whend(z*(t0),O+(z0,t0))<η,limt→∞d(z*(t),O+(z0,t0))=0.Next, we consider the following autonomous model with pulse effects:(34)dxdt=f(x,y),dydt=g(x,y),φ(x,y)≠0,Δx=ξ(x,y),Δy=η(x,y),φ(x,y)=0,
wheref(x,y)andg(x,y)are continuous differential functions defined onℝ2andφ(x,y)is a sufficiently smooth function with∇φ(x,y)≠0. Let(μ(t),ν(t))be a positiveT-periodic solution of model (34). The following result comes from Corollary2of Theorem1in [31].Lemma 12 (analogue of Poincaré criterion).
If the Floquet multiplierμsatisfies condition|μ|<1, where
(35)μ=∏j=1n‍κjexp{∫0T‍[∂f(μ(t),ν(t))∂x+∂g(μ(t),ν(t))∂y]dt}
with
(36)κj=((∂η∂y∂φ∂x-∂η∂x∂φ∂y+∂φ∂x)f++(∂ξ∂x∂φ∂y-∂ξ∂y∂φ∂x+∂φ∂y)g+)×(∂φ∂xf+∂φ∂yg)-1
andf,g,∂ξ/∂x,∂ξ/∂y,∂η/∂x,∂η/∂y,∂φ/∂x, and∂φ/∂yhave been calculated at the point(μ(τj),ν(τj)),f+=f(μ(τj+),ν(τj+)),g+=g(μ(τj+),ν(τj+)), andτj(j∈N)is the time of thejth jump, then,(μ(t),ν(t))is orbitally asymptotically stable.


## 4.2. Main Results

On the existence of positive order-1 periodic solution for model (30), we have the following theorem.Theorem 13.
For anyθ∈(0,1)andEX∈(0,1), model (30) admits a positive order-1 periodic solution.Proof.
Let pointA0((1-θ)EX,s^0)∈Γθfor sufficiently smalls^0withs^0≤θEX. In view of the geometrical structure of the phase space of model (30), trajectoryO+(A0,t0)of model (30) starts from initial pointA0and intersects sectionΓEat pointB1(EX,s1). And then, trajectoryO+(A0,t0)jumps to pointA1(s^1,(1-θ)EX)on sectionΓθdue to pulse control effectsx(t+)=(1-θ)x(t)andb(t+)=b(t)+θx(t). Sinces^0≤θEX, it follows that pointA1is above pointA0. Furthermore, pointB2is above pointB1. Otherwise, this is a contradiction with the uniqueness of solution for model (30). By (33), we have thats2=F(s1,θ,EX)and
(37)F(s1,θ,EX)-s1=s2-s1>0.
On the other hand, letλ0=1-(1-θ)EX. For critical pointC0((1-θ)EX,λ0), trajectoryO+(C0,t0)starts from initial pointC0; it intersects sectionΓEat pointD1(EX,λ1), then jumps to pointC1((1-m)EX,λ1+)on sectionΓθ, and finally reaches pointD2(EX,λ2)on sectionΓEagain. Sincex+b≤1, one has that(1-m)EX+λ1+≤1=(1-θ)EX+λ0. That is,λ1+≤λ0. Ifλ1+=λ0, namely, pointsC0andC1coincide, then model (30) has a positive order-1 periodic solution. Otherwise,λ1+<λ0; that is, pointC0is above pointC1, and pointD1is above pointD2. By (33), we can get thatλ2=F(λ1,θ,EX)and
(38)F(λ1,θ,EX)-λ1=λ2-λ1<0.
This together with (37) yields that Poincaré map (33) has a fixed point; that is, model (30) has a positive order-1 periodic solution. This completes the proof.Now, on the orbital stability of positive order-1 periodic solution of model (30), we have the following result.Theorem 14.
Let(φ(t),ψ(t))be a positive order-1 periodic solution of model (30) with periodT. For anyθ∈(0,1)andEX∈(0,1), if the Floquet multiplierμsatisfies condition|μ|<1, where
(39)|μ|=|κ|exp{∫0T‍[c(1-φ(t))(3s+k-ks-2)-cs(φ(t)+ψ(t))-csψ(t)]dt∫0T‍[c(1-φ(t))(3s+k-ks-2)}
with
(40)κ=(1-k)s(EX+ψ(T))-(1-θ)EX(1-s)(1-k)s(EX+ψ(T))-EX(1-s),
then(φ(t),ψ(t))is orbitally asymptotically stable.Proof.
Suppose that(φ(t),ψ(t))intersects sectionsΓθandΓEat pointsC+((1-θ)EX,θEX+ψ(T))andC(EX,ψ(T)), respectively. Comparing with model (34), we have that
(41)f(x,b)=c(1-x)[(1-k)s(x+b)-x(1-s)],g(x,b)=c(1-x)[kx-b(1-s-k)]-bcs(x+b),
andξ(x,b)=-θx,η(x,b)=θx,ϕ(S,I)=x-EX,(φ(T),ψ(T))=(EX,ψ(T)), and(φ(T+),ψ(T+))=((1-θ)EX,ψ(T)+θEX). Thus,
(42)∂f∂x=-c[(1-k)s(x+b)-x(1-s)]+c(1-x)(2s-ks-1),∂g∂b=-c(1-x)(1-s-k)-cs(x+b)-csb,∂ξ∂x=-θ,∂η∂x=θ,∂ϕ∂x=1,∂ξ∂b=∂η∂b=∂ϕ∂b=0.
Furthermore, it follows from (42) that
(43)κ=((∂η∂b∂ϕ∂x-∂η∂x∂ϕ∂b+∂ϕ∂x)f++(∂ξ∂x∂ϕ∂b-∂ξ∂b∂ϕ∂x+∂ϕ∂b)g+)×(∂ϕ∂xf+∂ϕ∂bg)-1=f+(φ(T+),ψ(T+))f(φ(T),ψ(T))=((1-(1-θ)EX)[(1-k)s(EX+ψ(T))-(1-θ)EX(1-s)](1-(1-θ)EX)[(1-k)s(EX+ψ(T)))×((1-EX)[(1-k)s(EX+ψ(T))-EX(1-s)(1-k)s(EX](1-EX)[(1-k)s(EX)-1,(44)μ=κexp{∫0T[-c((1-k)s(φ(t)+ψ(t))‍-φ(t)(1-s))+c(1-φ(t))×(2s-ks-1)-c(1-φ(t))(1-s-k)-cs(φ(t)+ψ(t))-csψ(t)]dt∫0T[-c((1-k)s(φ(t)+ψ(t))‍}.
On the other hand, integrating both sides of the first equation of model (30) along the orbitC+C^, we have that
(45)ln1-EX1-(1-θ)EX=∫(1-θ)EXEX‍dx1-x=-∫0T‍c[(1-k)s(φ(t)+ψ(t))-φ(t)(1-s)]dt.
From (43)–(45), we can obtain that
(46)|μ|=|(1-k)s(EX+ψ(T))-(1-θ)EX(1-s)(1-k)s(EX+ψ(T))-EX(1-s)|×exp{∫0T‍[c(1-φ(t))(3s+k-ks-2)-cs(φ(t)+ψ(t))-csψ(t)]dt∫0T‍}.
By condition (39), we see that model (30) satisfies all conditions of Lemma 12. Therefore, order-1 periodic solution(φ(t),ψ(t))of model (30) is orbitally asymptotically stable and has asymptotic phase property. This completes the proof.Remark 15.
From Theorem14, 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).Theorem 16.
For anyθ∈(0,1)andEX∈(0,1), model (30) has a positive order-1 periodic solution which is orbitally asymptotically stable.Proof.
Letλ*=1-(1-θ)EX, and suppose that the trajectoryO+(C0,t0)of model (30) starts from critical pointC0((1-θ)EX,λ*)and intersects sectionΓEat pointC*(EX,β*). In view of the geometrical construction of phase space of model (30) and that(1,0)is a globally asymptotically stable node, we obtain that trajectory of model (30) which starts from point((1-θ)EX,b)withb∈(0,λ*)will intersect sectionΓEat point(EX,b^); then,b^∈(0,β*). So we only need to consider trajectories of model (30) which start from the point(EX,b)on sectionΓE, whereb∈(0,β*).
Suppose that trajectoryO+(D1,t0)of model (30) which starts from initial pointD1(EX,b1)(0<b1<β*) jumps to pointD1+((1-θ)EX,b^1)on sectionΓθdue to pulse effects and then reaches sectionΓEat pointD2(EX,b2), whereb^1∈(0,λ*)andb2∈(0,β*). Repeating the above process, we have pulse points sequences{Dn(EX,bn)}and{Dn+((1-θ)EX,b^n)}, wherebn∈(0,β*)andb^n∈(0,λ*). Furthermore, this follows from Poincaré map (33) thatbn+1=F(bn,θ,EX)(n=1,2,…). On the other hand, for any two pointsDi(EX,bi)andDj(EX,bj)on sectionΓE, wherebi,bj∈(0,β*)andbi<bj, in view of pulse effects, pointDi+((1-θ)EX,b^i)is below pointDj+((1-θ)EX,b^j). And then, trajectoriesO+(Di,t0)andO+(Dj,t0)intersect sectionΓEat pointsDi+1(EX,bi+1)andDj+1(EX,bj+1), respectively. Therefore, from the geometrical construction of the phase space of model (30), we claim that
(47)0<bi+1<bj+1<β*.
In fact, if inequality (47) does not hold, that is,bi+1≥bj+1, then, it follows that pointDi+1is above pointDj+1or that two points coincide. Furthermore, we can obtain that trajectoriesO+(Di,t0)andO+(Dj,t0)intersect at a pointD*(x^,b^). It is indicated that there are two different solutions which start from pointD*. This is a contradiction with the uniqueness of solution of model (30). So inequality (47) is thus valid.
Now, for anyb0∈(0,β*), from Poincaré map (33) of sectionΓE, we haveb1=F(b0,θ,EX),b2=F(b1,θ,EX), andbn+1=F(bn,θ,EX)(n=2,3,4,…). In particular, ifb0=b1, then model (30) has a positive order-1 periodic solution, ifb0≠b1, then it follows from the fact (47) thatb0≠b2.
Next, we discuss the general circumstance; that is,b0≠b1≠b2≠⋯≠bk(k>2).(a)
Ifb0<b1, from (47), we obtain thatb1<b2. Then,b0<b1<b2. In this case, repeating the above process, we have that
(48)0<b0<b1<b2<b3⋯<β*.(b)
Ifb0>b1, similar to (a), we have that
(49)0<⋯<b3<b2<b1<b0<β*.
Therefore, in case (a), we can get thatlimn→∞bn=η. This implies that model (30) has an orbitally asymptotically stable positive order-1 periodic solution. Similarly, we can obtain the same result in case (b). This completes the proof.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 languagesXandYare coexistent and have a stable equilibrium state under state-dependent impulsive control strategy.Remark 18.
Similar results can also be obtained from the case thats<(1-s)(1-k); we, hence, omit them here.


## 4.3. 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 parametersc=k=0.5ands=0.7and control parametersEX=0.8andθ=0.15. From Figure 4, we see that state-dependent control measure plays an important role in preventing the disappearance of endangered language. Under state-dependent impulsive control measure, the downward trend for languageYwas controlled effectively, and the fractions of languageXand bilingualBare kept within reasonable levels. Furthermore, numerical simulations also show that the fractions of languagesX,Yand bilingualB, 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.The contrast of the dynamical behaviors of model (29) with state-dependent impulsive control measure and without control, wherec=k=0.5,s=0.7,EX=0.8, andθ=0.15.
(a)
(b)
(c)The orbitally asymptotical stability of positive order-1 periodic solution for model (29).
(a)
(b)
(c)Next, we investigate what effect has the choice of controlling parameters on the dynamical behavior of model (29) using numerical modeling method. We first chooseEX=0.8and parameterθto be0.05,0.15,0.25, and0.35, respectively. From Figure 6, we note that the length of times intervals between two control strategies are closely geared to the strength of control measureθand that the time interval increases with the increasing ofθ. Again from Figures 6(b) and 6(c), it is obvious that the fractions of languageYand bilingualBin population could maintain higher level for a long time due to largerθ. Of course, the cost of control measure is related to its strength. Furthermore, similar results can also be obtained fromθ=0.15and lettingEXbe0.9,0.85,0.8, and0.7in 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 languageYand bilingualBare kept at reasonable levels with the minimal cost of control measure? It is an interesting problem; at the same time, it is extremely difficult.The contrast of the dynamical behaviors of model (29) with different state-dependent impulsive control parameters, whereEX=0.8andθ=0.05, 0.15, 0.25, and 0.35, respectively.
(a)
(b)
(c)The contrast of the dynamical behaviors of model (29) with different state-dependent impulsive control parameters, whereθ=0.15andEX=0.9, 0.85, 0.8, and 0.75, respectively.
(a)
(b)
(c)


## 5. Concluding Remarks

The dynamic behavior of two-language competitive model (1) with parameters (2) andα=1is 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 languagesXandYare coexistent by adjusting the values of model parameterskands.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 of bilingualB, and some comparisons between with and without control are made in the figures.Finally, the dynamic behavior of model (1) with state-dependent 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 languagesX,Yand bilingualBat reasonable levels by adjusting control parameters. Theoretical basis for finding a new measure to protect the endangered language is provided.



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*Source: 654619-2013-12-03.xml*