#### 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.

#### 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: where capital lettersanddenote the two languages spoken in population; the uppercased letterdenotes the group of bilingual speakers; and the lowercased letters,, and(with) refer to the fraction of speakers of each of the languages in population and the fraction of bilingual speakers, respectively; parameterdenotes the probability of a monolingual speaker of languagebeing 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 bythe relative status of languageand bythat of language, whereis a normalization factor related to the time scale,is the power parameter, andis 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 [17–23] 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 account, we get the following reduced model (here, we assume thatthroughout the rest of this paper):

By the biological background of model (3), we only consider model (3) in the biological meaning region.

The following theorem is on the nonnegativity of solution of model (3).

Theorem 1. *The regionis positively invariant for model (3).*

The proof of Theorem 1 is simple; we, therefore, omit it here.

Now, we discussed the existence and stability of equilibria for model (3). The isocline corresponding tois the line. The isocline corresponding tois the line. Note that two isoclines do not have positive intersection point fororand have only one positive intersection point for. Namely, model (3) has only two trivial equilibriaandfororand has two trivial equilibria;and a positive equilibriumfor, where

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

Computing Jacobian matrixes from model (3) around,, and, we have that

From (6), sinceimplies that, it follows that the two eigenvalues ofare negative and that one eigenvalue is positive and another one is negative forwhen. ?Therefore, model (3) has a locally asymptotically stable nodeand a saddle pointfor. Furthermore, since there is not equilibrium in interior offor, model (3) has no periodic orbit lying entirely in the interior of. Therefore,is globally asymptotically stable in the interior offor this case. Similarly, sinceimplies that, we also have thatis globally asymptotically stable in the interior ofandis a saddle point for model (3) when.

Consider the eigenvalues of; letandbe the trace and determinant of, respectively. The eigenvalues ofare provided by the following characteristic equation:

Note that when,

Hence, both eigenvalues have negative real part, and, hence,is locally asymptotically stable when. Furthermore, when we mention that model (3) has no periodic orbits, refer to Lemmaof [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 any, one of the following statements is valid.*(a)*If, then model (3) has only two trivial equilibriaand, whereis a saddle point andis globally asymptotically stable in the interior of.*(b)*If, then model (3) has only two trivial equilibriaand, whereis a saddle point andis globally asymptotically stable in the interior of.*(c)*If, then model (3) has two saddle points,and a globally asymptotically stable positive equilibrium.*

*Remark 3. *From Theorem 2 and the equivalence of models (1) and (3), we obtain that model (1) has a globally asymptotically stable nodeand a saddle pointforand has a saddle pointand a globally asymptotically stable nodefor. Furthermore, if, model (1) has a coexistent equilibriumwhich is globally asymptotically stable.

Finally, we fix all parameters including,,, andand carry out numerical investigations to confirm our main results obtained in this section. Firstly, we choose,,, and; it is easy to calculate that. So, from the first conclusion of Theorem 2, we know that model (1) has only two trivial equilibriaand,is a globally asymptotically stable node, andis a saddle point, which is shown in Figure 1(a). It is clear that languageis permanent and languagewill fade away in this case. Similar conclusion can be obtained from Figure 1(b) with parameters,,, and. However, if we chooseand, then languagesandare coexistent and tend to positive equilibrium as shown in Figure 1(c).

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#### 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 Theorem 2 in Section 2, we know thatlanguageis permanent and languageand bilingual speakersare extinct forand thatlanguageis permanent and languageand bilingual speakersare extinct for. Considering the similarities of the two cases, we only need to consider case; that is,.

##### 3.1. Protecting of Language????with Continuous Control for??

We consider model (1) with parameters (2) and continuous control measure. The control system is modeled by the following differential equations: whereis a controlled variable, which means that the fraction of languagebecomes the bilingualper unit of time. Note that; model (10) can be written as follows:

We denote the right-side of model (11) byand, respectively. We, here, discuss the existence and asymptotical stability of positive equilibrium of model (11).

From, it is easily shown that

Substituting it in, it follows thatand due to conditionsand. Hence, there is at least onesuch that. Furthermore, it is also easy to calculate that for all. This together with (14) gives that for all, where inequalityis used. Hence,is unique. That is, model (11) has a unique positive equilibriumin the interior of.

Similar to the discussion of Theorem 2, we can get the locally asymptotical stability of equilibrium. Next, we discuss the global behavior of equilibrium. Let; we have So model (11) has no closed orbit lying entirely in the interior of. We, therefore, can show from the above discussion that equilibriumis globally asymptotically stable in the interior of. The result can be written by the following theorem.

Theorem 4. *For any, if, then model (11) has a globally asymptotically stable positive equilibriumin the interior of.*

*Remark 5. *From Theorem 2, it follows that languageis permanent and that languageand bilingual speakerswill eventually disappear forin model (1). However, if we introduce a control variable(no matter how small it is) in model (1), languagesandare coexistent and tend to positive equilibrium. The coexistent state, of course, depends upon the controlled strength. This implies thatis 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 variable; here,indicates an admissible control. In this optimal problem, we assume a restriction on the control variablesuch that, because conversion of all of languageat one time is impossible. In case of no control, the fraction of languageincreases while the fractions of languageand bilingualdie out. Therefore, the biological meaning of an optimal control in this problem is that the adequate levels for the fractions of languageand bilingualare 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 constantsandbalance the size of the termsand. Our goal is maximizing the fraction of bilingualand minimizing the systemic cost to the control measure. This seeks an optimal controlsuch that subject to model (10). It is obvious that the integrand of objective functionalis a convex function of control variableand 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 Hamiltonianwith respect toas follows: where,, andare adjoint variables.

In the following theorem, we derive the necessary conditions for the optimal control problem.

Theorem 6. *Letbe an optimal state solution with associated optimal control variablefor the maximized object functionalsubject to control model (10). Then, for model (10), there exist adjoint variables,, andsuch that
**
with transversality conditions,. Furthermore, the optimal control is given by
*

*Proof. *From Pontryagin’s Maximum Principle, adjoint variablescan be written as

These are just differential equations (21)–(23) with transversality conditions. 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.

##### 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,and. Besides that, we choose control variable. Figure 2(a), however, is completely different from Figure 1(a), which shows that model (10) has a globally asymptotically stable positive equilibrium. Furthermore, from the discussion of Theorem 4, it follows that valuesandwill 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.

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In addition, in Figure 2(c), it is interesting to note that control strengthis close to 1 (where), but languageruns around 0.2. This means that, no matter how strong the control strengthis, languagewill 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,, andin 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 bilingualin model (10) without and with continuous control, respectively. We see that the fractions of languageand bilingualin 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 languageslowly falls, the fraction of languagedecreases quite a lot, and the fraction of bilingualis 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 controlis plotted as a function of time for three different values of weight factor:,, and. The control variablefor the associated weight factoris much larger than the other two values. Note that, in general, asdecreases, the amount ofincreases. The same results can also be obtained from the expression ofin (27). The associated weight factoralso plays a significant role in keeping the balance of the size of fraction an optimal problem.

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#### 4. Protection of Language????with Impulsive Control for??

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:

The meaning of model (29) is as follows: when the fraction of languagereaches the critical threshold valueat time, controlling measure (for example, encouraging some speaker of languageto study language) is taken and the fractions of languageand bilingual speakersimmediately becomeand, respectively.

*Remark 7. *It is obvious that the fractions of languageand bilingualare rather small and in danger of becoming extinct when the fraction of languagereaches 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 valuerepresents the fraction of monolingual speakersin population, andrepresents 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 size,

By the biological background of model (30), we only consider model in the biological meaning region. 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

Suppose that pointis on section; then, trajectory jumps to pointon sectiondue to pulse effectsand, then intersects sectionat point, and finally jumps to pointon sectionagain, whereis decided byand parameters,. Therefore, we defined a Poincaré map of sectionas follows:

Letbe a solution of model (30) with initial condition. Trajectorystarts from point; first, it intersects sectionat point, then jumps to pointon sectiondue to pulse effects, and finally reaches pointon sectionagain. Repeating the above process, we have two points’ sequencesand. We notice that the coordinates satisfy the relation.

Letbe an arbitrary set in, and letbe an arbitrary point in. The distance between pointand setis defined by. For the convenience of statement in the rest of this paper, we introduce some definitions.

*Definition 9. *Trajectoryof model (30) is said to be order-periodic if there exists a positive integersuch thatis the smallest integer for.

*Definition 10 (orbital stability [30]). *Trajectoryis said to be orbitally stable if, given that, there exists a constantsuch that, for other solutionof model (30), when, one has thatfor all.

*Definition 11 (orbitally asymptotical stability [30]). *Trajectoryis said to be orbitally asymptotically stable if it is orbitally stable, and there exists a constantsuch that, for any other solutionof model (30), when,.

Next, we consider the following autonomous model with pulse effects: whereandare continuous differential functions defined onandis a sufficiently smooth function with. Letbe a positive-periodic solution of model (34). The following result comes from Corollaryof Theoremin [31].

Lemma 12 (analogue of Poincaré criterion). *If the Floquet multipliersatisfies condition, where
**
with
**
and,,,,,,, andhave been calculated at the point,,, andis the time of theth jump, then,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 anyand, model (30) admits a positive order-1 periodic solution.*

*Proof. *Let pointfor sufficiently smallwith. In view of the geometrical structure of the phase space of model (30), trajectoryof model (30) starts from initial pointand intersects sectionat point. And then, trajectoryjumps to pointon sectiondue to pulse control effectsand. Since, it follows that pointis above point. Furthermore, pointis above point. Otherwise, this is a contradiction with the uniqueness of solution for model (30). By (33), we have thatand

On the other hand, let. For critical point, trajectorystarts from initial point; it intersects sectionat point, then jumps to pointon section, and finally reaches pointon sectionagain. Since, one has that. That is,. If, namely, pointsandcoincide, then model (30) has a positive order-1 periodic solution. Otherwise,; that is, pointis above point, and pointis above point. By (33), we can get thatand

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. *Letbe a positive order-1 periodic solution of model (30) with period. For anyand, if the Floquet multipliersatisfies condition, where
**
with
**
thenis orbitally asymptotically stable.*

*Proof. *Suppose thatintersects sectionsandat pointsand, respectively. Comparing with model (34), we have that
and,,,, and. Thus,

Furthermore, it follows from (42) that

On the other hand, integrating both sides of the first equation of model (30) along the orbit, we have that

From (43)–(45), we can obtain that

By condition (39), we see that model (30) satisfies all conditions of Lemma 12. Therefore, order-1 periodic solutionof model (30) is orbitally asymptotically stable and has asymptotic phase property. This completes the proof.

*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).

Theorem 16. *For anyand, model (30) has a positive order-1 periodic solution which is orbitally asymptotically stable.*

*Proof. *Let, and suppose that the trajectoryof model (30) starts from critical pointand intersects sectionat point. In view of the geometrical construction of phase space of model (30) and thatis a globally asymptotically stable node, we obtain that trajectory of model (30) which starts from pointwithwill intersect sectionat point; then,. So we only need to consider trajectories of model (30) which start from the pointon section, where.

Suppose that trajectoryof model (30) which starts from initial point() jumps to pointon sectiondue to pulse effects and then reaches sectionat point, whereand. Repeating the above process, we have pulse points sequencesand, whereand. Furthermore, this follows from Poincaré map (33) that(). On the other hand, for any two pointsandon section, where,and, in view of pulse effects, pointis below point. And then, trajectoriesandintersect sectionat pointsand, respectively. Therefore, from the geometrical construction of the phase space of model (30), we claim that

In fact, if inequality (47) does not hold, that is,, then, it follows that pointis above pointor that two points coincide. Furthermore, we can obtain that trajectoriesandintersect at a point. It is indicated that there are two different solutions which start from point. This is a contradiction with the uniqueness of solution of model (30). So inequality (47) is thus valid.

Now, for any, from Poincaré map (33) of section, we have,, and. In particular, if, then model (30) has a positive order-1 periodic solution, if, then it follows from the fact (47) that.

Next, we discuss the general circumstance; that is,.(a)If, from (47), we obtain that. Then,. In this case, repeating the above process, we have that
(b)If, similar to (a), we have that

Therefore, in case (a), we can get that. 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 languagesandare coexistent and have a stable equilibrium state under state-dependent impulsive control strategy.

*Remark 18. *Similar results can also be obtained from the case that; 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 parametersandand control parametersand. 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 languagewas controlled effectively, and the fractions of languageand bilingualare 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.

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Next, we investigate what effect has the choice of controlling parameters on the dynamical behavior of model (29) using numerical modeling method. We first chooseand parameterto be,,, and, respectively. From Figure 6, we note that the length of times intervals between two control strategies are closely geared to the strength of control measureand that the time interval increases with the increasing of. Again from Figures 6(b) and 6(c), it is obvious that the fractions of languageand bilingualin 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 fromand lettingbe,,, andin 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 languageand bilingualare kept at reasonable levels with the minimal cost of control measure? It is an interesting problem; at the same time, it is extremely difficult.

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#### 5. Concluding Remarks

The dynamic behavior of two-language competitive model (1) with parameters (2) andis 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 languagesandare coexistent by adjusting the values of model parametersand.

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 bilingual, 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 languages,and bilingualat reasonable levels by adjusting control parameters. Theoretical basis for finding a new measure to protect the endangered language is provided.

#### Acknowledgments

The authors wish to thank the reviewers and the handling editor, Prof. Bashir Ahmad, for their comments and suggestions, which led to a great improvement to the presentation of this work. This paper has been partially supported by the Scientific Research Programmes of Colleges in Xinjiang (Grant no. XJEDU2011S08), the Natural Science Foundation of Xinjiang (Grant no. 2011211B08), the National Natural Science Foundation of China (Grants nos. 11001235, 11271312, and 11261056), the China Postdoctoral Science Foundation (Grants nos. 20110491750 and 2012T50836), and the Ministerio de Economia y Competitividad (Spain), Project no. MTM2010-15314 and has been co-financed by the European Community Fund FEDER.