1 Modelling Endngered Lnguges: The Effects of Bilingulism nd Socil Structure Jmes W. Minett,* nd Willim S-Y. Wng Lnguge Engineering Lbortory, Deprtment of Electronic Engineering, The Chinese University of Hong Kong, Shtin, Hong Kong Abstrct The mthemticl model for lnguge competition developed by Abrms nd Strogtz llows the evolution of the numbers of monolingul spekers of two competing lnguges to be estimted. In this pper, we extend the model to exmine the role of bilingulism nd socil structure, neither of which re ddressed in the previous model. We consider the impct of two strtegies for lnguge mintennce: 1) incresing the sttus of the endngered lnguge; nd 2) djusting the vilbility of monolingul nd bilingul eductionl resources. The model llows us to predict for which scenrios of intervention lnguge mintennce is more likely to be chieved. Qulittive nlysis of the model indictes set of intervention strtegies by which the likelihood of successful mintennce cn be incresed. Keywords: lnguge competition; lnguge deth; lnguge mintennce; dynmicl system; gentbsed model. This is pre-print of n rticle ccepted for publiction by Lingu. * Corresponding uthor: J. W. Minett. Tel.: +852 3163 4346; fx: +852 2603 5558. E-mil ddresses: jminett@ee.cuhk.edu.hk (J. W. Minett), wsywng@ee.cuhk.edu.hk (W. S-Y. Wng).
2 1. Introduction The 6,000 or so lnguges spoken on our plnet tody re the products of numerous millenni of culturl evolution. They encpsulte the experience nd knowledge of diverse peoples collected in widely different environments, nd re precious prt of the humn heritge. With the explosive expnsion of few dominnt lnguges in recent decdes, t lest hlf of the world s lnguges re criticlly endngered in tht they will soon hve no spekers nd become extinct (Kruss, 1992; Crystl, 2000). Pgel (1995) estimtes tht roughly 140,000 lnguges hve ever existed (medin estimte), so it is the fte of the mjority of lnguges to become extinct. Fishmn (1991) rgues tht deth of lnguge generlly leds to deth of the underlying culture to which it is linked. It is therefore n importnt chllenge to understnd such situtions s precisely s possible, nd to recognize whether there re mesures tht cn help us preserve some of this heritge. Much work hs been crried out on both theoreticl nd empiricl issues of chieving lnguge mintennce s evidenced by the numerous recent volumes on the subject (e.g. Fishmn, 1991; Grenoble & Whley, 1998; Crystl, 2000; Nettle & Romine, 2000; Fishmn, 2001; Brdley & Brdley, 2002; Grenoble & Whley, 2006). Fishmn (2001:1) begins his tretise by stting: Wht the smller nd weker lnguges (nd peoples nd cultures) of the world need re not generlized predictions of dire nd even terminl illnesses but, rther, the development of therpeutic understndings nd pproches tht cn be djusted so s to tckle essentilly the sme illness in ptient fter ptient. Towrd this end, Abrms nd Strogtz (2003) hve proposed mthemticl model for studying lnguge competition. The model predicts tht whenever two lnguges compete for spekers, one lnguge will eventully become extinct, the lnguge tht dies depending on the initil proportions of spekers of ech lnguge nd their reltive sttus. The model obtins good fit to number of empiricl dt sets, trcing the reltive bundnce of spekers of severl
3 endngered lnguges tht hve been undergoing competition with other, more prestigious lnguges. However, it does not ccount for either bilingul individuls or the socil structure of the popultion within which the lnguges compete. The model lso does not distinguish the verticl nd horizontl trnsmission of lnguge, nd ignores the impct of the behviour of individul spekers in the popultion, focusing insted on the expected ggregte behviour of the popultion s whole. Despite these limittions, the model hs stimulted burst of reserch into the dynmics of lnguge competition nd diversity (e.g., Ptrirc & Leppänen, 2004; Mir & Predes, 2005; Wng & Minett, 2005; Schulze & Stuffer, 2006; Stuffer, Cstelló, Eguíluz & Sn Miguel, 2006), much of it performed by non-linguists. In this pper, we extend the Abrms nd Strogtz work to model bilingulism explicitly, ccounting for the fct tht some individuls my spek both of the competing lnguges. Our first step is to formulte mthemticl model tht, like the Abrms nd Strogtz model, dels with the expected ggregte behviour of the whole popultion this model lso predicts tht deth of one of the two competing lnguges is inevitble (lthough the trjectories tht led to this stte differ from those of the previous model). However, in order to investigte the rnge of possible trjectories tht system of two competing lnguges cn potentilly follow from the sme initil stte, nd to devise method by which the probbilities ssocited with them cn be predicted, we lso implement n gent-bsed model. In prticulr, we investigte the impct of simple strtegies for lnguge mintennce, llowing us to estimte for different scenrios the reltive likelihood tht pir of competing lnguges cn be mintined in popultion comprising both monolingul spekers of ech lnguge nd bilingul spekers. We then exmine the role of socil structure on the probbilities of mintennce, representing the socil structure by locl-world network (Li & Chen, 2003) to encpsulte the ptterns of sociolinguistic interctions mong the individuls comprising the popultion.
4 The pper is lid out s follows: In Section 2, we discuss deterministic models of lnguge competition, first briefly describing the Abrms nd Strogtz model for popultion in which two lnguges compete for spekers, then introducing n extension of the model tht incorportes modelling of bilingulism. In Section 3, we pply the extended model to investigte the efficcy of simple strtegies for lnguge mintennce. The deterministic models tht we discuss in Sections 2 nd 3 llow us to determine the expected finl stte of system in which two lnguges compete for spekers. However, they do not revel the vriety of competing outcomes tht re possible, the likelihood tht the system converges to ech of them, or the impct of popultion size on the likelihood of mintennce. Therefore, in Section 4 we go on to derive n gent-bsed model for lnguge competition tht llows such effects to be ssessed. Our simultions suggest some generl principles for chieving mintennce. We then consider the impct of socil structure on the probbility of successful mintennce in Section 5. The pper concludes in Section 6. 2. Deterministic Models of Lnguge Competition 2.1 The Monolingul Model of Abrms nd Strogtz (2003) Abrms nd Strogtz (2003) (henceforth A&S) hve developed simple mthemticl model of lnguge competition for popultion in which two lnguges compete for spekers. The model ims to trce the vrition over time of the proportion of spekers of ech lnguge, potentilly llowing lnguge tht is endngered to be identified t n erly stge so tht pproprite ction to mintin it cn be plnned. In deriving their deterministic model, A&S mke number of simplifying ssumptions, in prticulr: i) Ech individul is monolingul in one of the two lnguges, but no individul speks both lnguges.
5 ii) Spekers of one lnguge switch to spek the second lnguge ccording to the ttrctiveness of tht second lnguge. iii) The ttrctiveness of lnguge increses with both the proportion of spekers of tht lnguge nd its sttus, globl prmeter tht mesures its inherent utility within the community we shll discuss sttus in Section 3. iv) The popultion hs uniform socil structure: individuls interct with ech other t the sme rte, nd influence the lnguges tht they ech use eqully we discuss socil structure in Section 5. v) The popultion size remins constnt. Let us consider how these ssumptions trnslte into mthemticl model. We write the proportions of monolingul spekers of X nd Y s x nd y, respectively. Since ech individul is ssumed to be monolingul in one of these two lnguge, the vlues of x nd y must sum to 1 (i.e., x + y = 1). The rte t which spekers of one lnguge switch to become spekers of the second lnguge depends on the ttrctiveness of tht second lnguge. In their most generl conception of ttrctiveness, A&S ssume tht lnguge hs greter ttrctiveness the more monolingul spekers it hs nd the greter its sttus. All such functionl forms of ttrctiveness, they stte, give rise to the sme qulittive dynmics s tht of the following constrined model in which the ttrctiveness of lnguge X to spekers of Y is given by the power-lw P YX = c s X x. (1) In the bove formul, s X denotes the sttus of lnguge X, nd is prmeter tht models how the ttrctiveness of X scles with the proportion of spekers of X. The ttrctiveness of Y to spekers of X cn be stted similrly. For = 1, the ttrctiveness of X increses linerly with its proportion of spekers. For greter 1, doubling the proportion of spekers more thn doubles the ttrctiveness, nd vice vers. The prmeter c in (1) indictes the pek rte t which spekers of Y switch to spek X. This cn reflect, vriously, the rte of contct between
6 pirs of individuls, the propensity for individuls to lern new lnguge bsed on their existing linguistic skills, or the provision of lnguge resources to children. The formul (1) therefore provides simple model for vriety of sociolinguistic fctors tht influence the ttrctiveness of lnguge, nd whose dynmics reflect those of the entire clss of models in which ttrctiveness increses monotoniclly with both the proportion of monolingul spekers nd sttus. In this model, Y monolinguls switch lnguge to become monolingul spekers of X t rte proportionl to P YX. Likewise, X monolinguls become Y monolinguls t rte proportionl to P XY (defined similrly to (1)). The rte t which the proportion of monolingul spekers of X chnges over time cn therefore be written symboliclly s dx dt = y P x. (2) YX P XY Substituting for P YX nd P XY in (2), we obtin the formul dx dt [ y x s x y s ] = c. (3) X y The rtes of chnge of the system re summrized in Figure 1. We cn mke further simplifictions to the formul by ssuming, s do A&S, tht s X + s Y = 1, nd by substituting (1 x) for y, to obtin dx dt [( x) x s x ( 1 x) ( s )] = c 1 1. (4) X X Becuse the proportions of spekers of X nd Y sum to 1, there is no need to consider seprtely the rte t which the proportion of Y monolinguls chnges with time this is just minus the rte of chnge of the X monolinguls.
7 The dynmics of the model cn be nlyzed by seeking the vlues of x t which no chnge in the proportion of X monolinguls is expected to tke plce. These points re termed equilibri, nd re of two types: stble nd unstble. 1 In order to locte the equilibri, we clculte the vlues of x for which the rte of chnge of x is zero. Stble equilibri occur t x = 0 nd t x = 1. A third equilibrium does occur for n intermedite vlue of x, corresponding to sitution in which spekers of both lnguges remin, but this equilibrium is unstble. Therefore, no mtter wht is the initil stte, the system will ultimtely end up t one of the two stble equilibri. The significnce for lnguge competition is tht the model implies tht one lnguge will lwys cquire ll the spekers in the popultion, cusing the lnguge with which it competes to become extinct. However, A&S suggest tht third stble equilibrium, corresponding to stte in which the two competing lnguges re both mintined, cn be chieved by pproprite control of the sttus of the endngered lnguge. The ide of controlling the sttus, s well s other model prmeters, in order to chieve mintennce forms the bsis of the lnguge mintennce strtegies tht we pursue in Section 3. A&S hve tested the ccurcy of their model by fitting it to dichronic dt collected for three endngered lnguges: Scottish Gelic, Welsh nd Quechu. The former two dt sets re bsed on census dt, while the ltter dt derive from records of the lnguges used in religious services in Peru. In ech cse, the set of vlues of the model prmeters for which the corresponding trjectory most closely mtches the dichronic dt cn be clculted. In this wy, the sttus of Scottish Gelic ws estimted to be 0.33, hlf tht of English with which it continues to compete. Welsh ws estimted to hve slightly higher sttus with respect to English, 0.4, but the predicted trjectory for the proportion of spekers of Welsh throughout 1 Informlly, n equilibrium, or fixed point, of dynmicl system is stte in which the system will remin, once ttined. In other words, n equilibrium is stte for which the rte of chnge is zero, i.e., dx/dt = 0. An equilibrium is sid to be stble when the system, upon smll perturbtion from the equilibrium, returns bck to tht stte. We shll refer to ll equilibri tht re not stble s unstble. More forml definitions of these nd other terms relting to dynmicl systems cn be found, for exmple, in Strogtz (1995).
8 Wles nevertheless shows strong downwrd trend tht fits the empiricl dt well. A&S lso estimted the vlue of prmeter for ech dt set, nd found to cluster bout the men vlue 1.31. This vlue of exceeds 1, indicting tht the ttrctiveness of lnguge more thn doubles s its proportion of spekers doubles. We will therefore ssume tht the prmeter tkes vlue no less thn 1 in the course of development of our own model. The model of A&S ppers to work well in modelling chnges in the ptterns of lnguge usge within popultion in which two lnguges compete. However, the model dels only with monolingul spekers. In prctice, we observe tht typiclly speker does not suddenly give up one lnguge completely in fvour of nother it is extremely rre, for exmple, for children to loose the bility to communicte with their prents. Almost lwys, spekers will mintin the lnguge cquired from their prents nd, perhps, lern dditionl lnguges to vrious degrees, prticulrly while young. Such spekers my switch lnguges bck nd forth, depending on the context, whether it be home, school, or workplce. The nture nd extent of the bilingulism will depend on vriety of societl fctors. We therefore believe tht the incorportion of bilingulism is essentil for relistic modelling of lnguge deth. 2.2 A Bilingul Model Incorporting Verticl nd Horizontl Trnsmission We here extend the A&S model by explicitly modelling bilingulism, which we ccommodte by introducing third clss of spekers, Z, who spek both X nd Y. Wheres A&S model only two types of trnsition (X Y nd Y X), there re six types of trnsition possible mong monolingul nd bilingul spekers of two lnguges (X Y, Y X, X Z, Z X, Y Z, nd Z Y). However, the trnsitions X Y nd Y X re exceedingly rre in prctice, whether s result of verticl trnsmission or horizontl trnsmission. One would not expect child of monolingul spekers of X, sy, to cquire only lnguge Y. Nor would one expect n dult who previously spoke only lnguge X to then cquire Y nd simultneously forget how to spek X. The other four types of trnsition, however, ll occur frequently in
9 prctice. We therefore model only trnsitions of the four types X Z, Z X, Y Z, nd Z Y, s suggested by Wng & Minett (2005). Consider first verticl trnsmission. Children of monolingul prents necessrily cquire the lnguge of their prents s their first lnguge. However, children of bilinguls my cquire either or both of the competing lnguges. For simplicity, we dopt uni-prentl model of trnsmission. We therefore dopt the following model for verticl trnsmission (V-Model): All children of monolingul prents cquire the lnguge of their prents, tht is X X nd Y Y; Children of bilingul prents cquire only lnguge X from their prents, tht is Z X, t rte proportionl to the ttrctiveness of X; Children of bilingul prents cquire only lnguge Y from their prents, tht is Z Y, t rte proportionl to the ttrctiveness of Y; All other children of bilingul prents cquire both lnguges from their prents so tht they too become bilingul, tht is Z Z. We ssume tht the ttrctiveness of lnguge X to child of bilingul prent increses with both the sttus of X nd the proportion of monolingul spekers of X. For given sttus, X is mximlly ttrctive when the entire popultion is monolingul in X, nd minimlly ttrctive when none of the popultion is monolingul in X. Given this, it follows tht the ttrctiveness of cquiring both lnguges to child of bilingul prent is mximl when the entire popultion is bilingul, nd miniml when the entire popultion (other thn the prent) is monolingul. Adopting the sme functionl form for ttrctiveness s in (1) bove, we write the ttrctiveness of being monolingul in X to children of bilingul prents s P ZX = czx sx x. (5)
10 A similr formul holds for the ttrctiveness of Y. Notice tht in (5) we hve defined prmeter, c ZX, which llows the pek ttrctiveness of X to be modelled independently from tht of Y, reflecting, for exmple, the different vilbility of eductionl resources in ech lnguge. Control of these prmeters will form the bsis of proposed strtegy for lnguge mintennce tht we discuss in Section 3. Symboliclly, the V-Model for lnguge competition cn be written s dx dt dy dt = = z P z P ZX ZY, (6) where z denotes the proportion of bilingul spekers. Note tht, for the bilingul model, x + y + z = 1. The rte of chnge of the proportion of bilinguls is therefore simply minus the sum of the rtes of chnge of the proportions of monolingul spekers of X nd Y. By substituting for P ZX nd P ZY in (6), we obtin the V-Model dx dt dy dt = = c c ZX ZY s s X Y ( 1 x y) x ( 1 x y) y. (7) Wht then of horizontl trnsmission? We hve ssumed tht dults retin sufficient knowledge of lnguge, once cquired, for tht lnguge to be vilble for trnsmission to ny offspring they might produce. Tht being the cse, bilingul dults re ssumed to remin bilingul throughout their lifetime. Monolingul dults, however, my either remin monolingul or subsequently become bilingul by cquiring the second lnguge. We therefore dopt the following model for horizontl trnsmission (H-Model): All bilingul dults remin bilingul, tht is Z Z; Adults speking X only subsequently cquire lnguge Y, tht is X Z, t rte proportionl to the ttrctiveness of Y;
11 Adults speking Y only subsequently cquire lnguge X, tht is Y Z, t rte proportionl to the ttrctiveness of X; All other monolingul dults remin monolingul, tht is X X nd Y Y. In ddition to being ble to communicte with both X monolinguls nd bilinguls, the X monolingul dult who subsequently cquires lnguge Y cn dditionlly communicte with Y monolinguls. Therefore, we ssume tht the ttrctiveness of cquiring lnguge Y, so becoming bilingul, to monolingul spekers of X increses with both the sttus of Y nd the proportion of monolingul spekers of Y: P XZ = c XZ sy y, (8) functionl form consistent with (5). The ttrctiveness, P YZ, of becoming bilingul to monolingul spekers of Y is defined similrly. The H-Model cn be written s dx dt dy dt = = x P y P XZ YZ, (9) which reduces to the following system fter substituting for P XZ nd P YZ : dx dt dy dt = = c c XZ YZ s s Y X x y. (10) y x To develop unified bilingul model for lnguge competition, encompssing both verticl nd horizontl trnsmission, we model the rtes t which individuls follow the V-Model nd H-Model. We do so by defining mortlity rte, µ, t which dults re replced by children. Children cquire lnguges from their prents ccording to the V-Model; surviving dults ll
12 cquire lnguges ccording to the H-Model. The unified bilingul model cn therefore be stted explicitly s dx dt dy dt = = µ c µ c ZX ZY s s X Y ( 1 x y) x ( 1 µ ) XZ ( 1 x y) y ( 1 µ ) c s y x c YZ s Y X x y. (11) The rtes of chnge of the unified bilingul model re summrized in Figure 2. For convenience, throughout the reminder of this pper, we will ssume, without loss of generlity, tht s X + s Y = 1, llowing us to substitute s Y with (1 s X ). In order to determine the dynmics of the unified bilingul model, nd thereby drw conclusions regrding the competition of two lnguges, we study the direction field 2 of the system (11) for different sets of vlues of the model prmeters s X, c XZ, c YZ, c ZX, nd c ZY. Figure 3 shows the direction field for two vlues of the sttus of lnguge X (s X = 0.5 & s X = 0.3). The system hs four equilibri: X* nd Y* correspond to ll individuls being monolingul in X nd Y, respectively; Z* corresponds to ll individuls being bilingul; nd U* corresponds to stte in which there re some spekers of ech type. It cn esily be shown (by linerizing (11) bout ech equilibrium) tht X* nd Y* re both stble, mening tht once the system hs pproched either of these two sttes, it will remin nerby. However, both Z* nd U* re unstble, mening tht even if these sttes re pproched, the system will subsequently tend to move wy from them. The position of U* shifts ccording to the vlues of the model prmeters. For X hving the sme sttus s Y (s X = 0.5, shown in pnel ), the position of U* corresponds to stte in which X nd Y hve the sme numbers of monolingul spekers. However, s the sttus of X is reduced, so the position of U* moves towrds the stble equilibrium X*. As result, fewer trjectories converge on X*, mening tht lnguge X will become extinct for greter rnge of 2 The direction field of dynmicl system is digrm tht shows, for representtive set of sttes of tht system, the direction in which it is expected to chnge stte, the direction of chnge being indicted by n rrow. A similr digrm, clled phse portrit, lso represents the rte of chnge t ech point by the length of the rrow (longer rrows indicte fster rte of chnge)
13 initil sttes. In terms of the lnguge competition, the model predicts tht one of the two competing lnguges will eventully cquire ll the spekers, regrdless of the initil conditions, resulting in monolingul system in which only one lnguge is spoken. 2.3 The Bilingul Model of Mir & Predes (2005) A different pproch to modelling bilingulism, in which the two lnguges tht compete for spekers re prtilly mutully intelligible, hs been suggested by Mir & Predes (2005). When two lnguges re prtilly mutul intelligible, monolingul spekers of one lnguge cn sometimes communicte effectively with monolingul spekers of the competing lnguge. Such communictions potentilly llow monolingul spekers to become bilingul. Monolingul spekers of lnguge X, sy, re therefore ssumed to become bilingul t rte tht is proportionl both to the proportion of spekers of the competing lnguge Y (s in our extension of the A&S model) nd to the degree of mutul intelligibility. Correspondingly, the greter the mutul intelligibility, the lesser is the proportion of monolingul spekers of X ttrcted to become monolingul in Y. Bsed on these mendments to the model, Mir & Predes derive the system dx dt dy dt = = c c [( 1 k)( 1 x)( 1 y) s x( 1 x) ( 1 s )] X [( 1 k)( 1 y)( 1 x) ( 1 s ) y ( 1 y) s ] x x X, (12) where k, with vlue between 0 nd 1, denotes the degree of mutul intelligibility between lnguges X nd Y. This system is noteworthy since, for sufficiently lrge k, stble equilibrium is introduced in which both monolingul nd bilingul spekers of the more prestigious lnguge survive. Although no monolingul spekers of the endngered lnguge remin, tht lnguge is preserved mong the bilingul members of the popultion. Although the model provides good fit to dt for the competition between the Cstilin nd Glicin dilects of Spnish during the period 1875 1975 (Mir & Predes, 2005), we doubt
14 tht the model cn be usefully pplied in its current form to generl situtions of lnguge competition. Often, competing lnguges re not mutully intelligible (k 0), in which cse the model reduces to the monolingul model of A&S, which, we hve rgued, is unsuited to modelling lnguge competition due to the lck of bilingul modelling. Also, even when the competing lnguges re mutully intelligible dilects or closely geneticlly relted lnguges, it is not cler tht individul spekers mke the trnsition directly between monolingul use of one lnguge to monolingul use of the competing lnguge (X Y or Y X) in ny but the rrest of situtions. Moreover, there is no linguistic dvntge conferred to bilingul individul when no monolingul spekers of both of the competing lnguges remin: in such sitution, one would expect the bilinguls to be quickly replced by monolinguls, resulting in the extinction of the endngered lnguge, behviour tht is reproduced by the model we hve proposed in Section 2.2. Nevertheless, tweking the model, by constrining the trnsitions tht re likely to occur in prctice, s dvocted in the previous section, nd by considering the impct of socil structure, s we will do in Section 5, my result in model tht llows the dynmics of competition between mutully intelligible, relted lnguges to be better understood. 3. Lnguge Mintennce Fishmn hs proposed the 8-stge Grded Intergenertionl Disruption Scle (GIDS) by which the prospects for the continuing usge of prticulr lnguge in community cn be ssessed (Fishmn, 1991; Fishmn 2001). The GIDS cn be used to identify the contexts in which the lnguge is spoken, rnging from ntionwide usge throughout the mss medi nd governmentl opertions, to the lnguge being tught s n option in litercy schools, to sprse usge by socilly isolted elderly people. By using this scle, lnguges tht re thretened with extinction cn be detected. Once lnguge hs been identified s endngered, it must be decided whether or not n ttempt will be mde to mintin it. It is well known tht mintennce of n endngered
15 lnguge cn sometimes be chieved by top down processes like legisltion or by bottom up movements like ethnic pride. Sometimes, however, the socio-economic fctors tht led to lnguge becoming endngered in the first plce might discourge its spekers nd linguists from seeking to mintin it. As Mufwene writes, Linguists concerned with rights of lnguges must sk themselves whether these rights previl over the rights of spekers to dpt competitively to their new socioeconomic ecologies (Mufwene 2004: 219). Our focus in this pper is not to dvocte lnguge mintennce in ll situtions. Rther, we im to develop quntittive models tht cn ssist in the identifiction of mintennce strtegies tht cn be pplied in certin situtions of lnguge endngerment. It is for the community whose indigenous lnguge is endngered to decide whether mintennce should be undertken. Crystl (2000) hs identified six min mechnisms of intervention by which mintennce my be ttempted: i) Incresing the prestige of its spekers. ii) iii) iv) Incresing the welth of its spekers. Incresing the power of its spekers. Improving its presence in the eductionl system. v) Ensuring tht the lnguge cn be written down. vi) Providing ccess to electronic technology to its spekers. The first three mechnisms relte to the sttus of the lnguge, globl ssessment of the socioeconomic dvntges conferred to members of the community speking tht lnguge. Bringing bout n increse in the sttus of lnguge my serve to mintin it. The ltter three mechnisms relte to people s ccess to the lnguge, the ese with which they cn come to mke productive use of the lnguge. For exmple, promoting the teching of n endngered lnguge t school my enhnce the fluency of its existing spekers nd encourge bilingulism
16 mong spekers of the lnguge with which it competes, both of which my hve impct on its vibility. 3.1 Modelling Lnguge Mintennce We model the effects of dopting such mechnisms of intervention s djustments to the prmeters s X, c XZ, c YZ, c ZX, nd c ZY of the system (11). In prticulr, we seek strtegies for controlling these prmeters such tht n endngered lnguge, X, nd its competitor, Y, cn both be mintined. We mke no ttempt here to determine optiml strtegies for lnguge mintennce there is little vlue in doing so since, t this time, the precise quntittive reltionships mong the vrious mintennce mechnisms nd the prmeters of models such s the one proposed here re still only poorly understood. Rther, we investigte conditions under which intervention tht brings bout significnt chnges to the model prmeters of n endngered lnguge cn led to tht lnguge being mintined. We suppose tht the vlue of n rbitrry prmeter, θ, of system (11) cn be represented s function, θ(x), of the proportion of monolingul spekers of the endngered lnguge, X. In order to obtin model tht is menble to nlysis, we ssume tht community cn bring bout chnge in the vlue of the prmeter whenever the proportion of monolingul spekers of X flls below some threshold, x < th X. Symboliclly, we write this s θ : x thx θ ( x) =, (13) θ : x < thx where θ nd θ re both constnts, θ representing the vlue of the prmeter prior to intervention, nd θ representing its vlue fter intervention. A grph of θ(x) is shown in Figure 4. In prctice, the extent to which community cn bring bout chnge in prmeter is likely to be limited, nd the speed with which it cn do so constrined. However, more relistic intervention model, hving less brupt onset nd offset (indicted in the figure by the dotted lines), gives rise to the sme qulittive behviour s (13), which we now describe.
17 We ssume tht the system (11) is vlid both pre- nd post- intervention, but with different set of vlues of the prmeters s X, c ZX, c ZY, c XZ, nd c YZ. Figure 5 illustrtes the impct of n intervention tht brings bout n increse in the sttus of lnguge. The first pnel shows the direction field for the system without intervention when the sttus of the lnguge X is 0.4. The only stble sttes correspond to deth of either lnguge X or lnguge Y. The second pnel shows the direction field corresponding to lnguge X hving sttus 0.6. Agin, the only stble sttes correspond to deth of one of the two competing lnguges. Pnel (c), however, shows the direction field for the system in which the pre-intervention sttus of X is incresed from 0.4 to the post-intervention sttus 0.6 whenever the proportion of monolingul spekers of X flls below 30%. By intervening in this wy, n dditionl stble fixed point, S*, is creted between the two unstble fixed points, U 1 * nd U 2 *. We cn better understnd which strtegies give rise to the stble fixed point S*, nd so led to the potentil mintennce of the two competing lnguges, by studying the nullclines of the system (11). The nullclines re lines long which the rte of chnge of either x or y is zero: tht is, dx/dt = 0 or dy/dt = 0. The system without intervention hs two nullclines, shown superimposed on the direction field in Figure 6. The nullclines intersect t the unstble equilibrium U* nd prtition the stte spce into four regions, ech of which is chrcterized by direction of chnge of x (incresing or decresing) nd direction of chnge of y (incresing or decresing), s shown in Figure 6b. Trjectories tht strt in two of these regions (R U nd R Z ) initilly move towrds the equilibrium U* but, becuse this equilibrium is unstble, then enter one of the two remining regions (R X nd R Y ) fter which the system converges to one of the two stble sttes, X* or Y*. We will shortly mke use of the fct tht the regions R X nd R Y re both trpping regions, i.e., no trjectory cn exit from them. The nullclines thus provide summry of the qulittive dynmics of the system. Suppose tht the proportion of monolingul spekers of n endngered lnguge X hving sttus 0.2 hs fllen to 30%, nd tht the community in which it is spoken is to intervene in
18 order to bring bout its mintennce. How should the community intervene nd wht is the corresponding effect on the nullclines? Figure 7 shows the direction field for the system without intervention. For x 0.3, lmost ll trjectories converge to Y* (i.e., lnguge X dies). Figure 7b shows the impct of incresing the sttus of X from 0.2 to 0.4 whenever the proportion of monolingul spekers of X flls below 30%. This djustment of the sttus of X leds to n increse in the slopes of both of the post-intervention nullclines, bringing bout chnge in the post-intervention dynmics. However, the post-intervention nullclines do not intersect, so no stble equilibrium is introduced. Thus, intervening to increse the sttus of the endngered lnguge from 0.2 to 0.4 fils to chieve mintennce. Figure 7c shows the impct of n lterntive strtegy for intervention: incresing the pek rte, c ZY, t which children of bilingul spekers cquire only lnguge Y. Incresing the vlue of this prmeter hs the effect of incresing the proportion of monolingul spekers of Y. For this strtegy, the slope of only one nullcline is incresed (dy/dt = 0). Agin, the post-intervention nullclines do not intersect, nd intervention fils to bring bout stble stte in which both lnguges re mintined. Figure 7d, however, shows the sitution in which both the sttus of X nd the pek trnsition rte, c YZ, re incresed. In this cse, the post-intervention nullclines re incresed in slope sufficiently tht they intersect t x < 0.3. Region R Y (see Figure 6b) of the pre-intervention stte spce buts region R X of the post-intervention stte spce. This is sufficient to introduce stble equilibrium, S*, in which both lnguges re spoken, t x = 0.3, y 0.15, z 0.55 trjectories to the left of S* lie in region R X of the post-intervention stte spce nd therefore pproch S* from the left, nd trjectories to the right of S* lie in region R Y of the pre-intervention stte spce nd therefore pproch S* from the right. The greter re the slopes of the nullclines, the lrger re the corresponding regions R X nd R Y, which, in turn, enlrges the bsin of ttrction of S*: the rnge of initil sttes for which the system converges to S*. This behviour points to set of generl principles tht cn guide strtegies for lnguge mintennce. Tble 1 lists set of mechnisms by which the slope of the post-intervention
19 nullclines cn be incresed in order tht they intersect, so enlrging the bsin of ttrction of the stble equilibrium, nd thereby incresing the likelihood of successful mintennce. Tble 1. Mechnisms to increse the slope of the post-intervention nullclines, nd thereby increse the likelihood of successful mintennce. incresing the post-intervention sttus, s X, of X increses the slopes of the postmintennce nullclines dx/dt = 0 or dy/dt = 0; incresing the post-intervention pek trnsition rte c ZX increses the slope of the post-mintennce nullcline dx/dt = 0; incresing the post-intervention pek trnsition rte c ZY increses the slope of the post-mintennce nullcline dy/dt = 0; decresing the post-intervention pek trnsition rte c XZ increses the slope of the post-mintennce nullcline dx/dt = 0; decresing the post-intervention pek trnsition rte c YZ increses the slope of the post-mintennce nullcline dy/dt = 0. Thus, for given intervention threshold, th X, the likelihood of successful mintennce is incresed by incresing the post-intervention vlues of prmeters s X, c ZX nd c ZY, nd by decresing the post-intervention vlues of prmeters c XZ nd c YZ. In other words, the sttus of the endngered lnguge should be incresed nd the two lnguges isolted by encourging monolingul eduction of children. 4. An Agent-bsed Model of Lnguge Competition We now exmine the dynmics of lnguge competition from different perspective, introducing n gent-bsed model tht llows us to estimte the likelihood tht the system converges to ech stble stte. The model tht we hve introduced in Sections 2 nd 3 is
20 deterministic: given the initil stte, its stte t ny future time is uniquely determined. The system llows, in principle, the proportion of spekers in ech stte to be clculted (even if one is not lwys ble to estblish n nlyticl expression for it). However, which prticulr spekers re bilingul t some time nd which of them produce monolingul offspring, for exmple, re not ddressed. The model does not trce the sttes of every single speker, only the proportions of spekers hving ech stte. Thus, it specifies model of the expected behviour of the competition, but not the rnge of behviours tht cn result from given initil stte or their reltive likelihoods. Typiclly, the behviour of n ppropritely defined system of differentil equtions pproches the ctul behviour of the underlying system being modelled when the popultion size is lrge. However, the popultion sizes tht re relevnt in the context of lnguge endngerment nd mintennce re often of the order of hundreds or thousnds of individuls. For such smll popultions, fluctutions in the lnguge usge ptterns of certin individuls my led to dynmics tht diverge significntly from the expected behviour. Even when we consider the mintennce of n endngered or minority lnguge hving considerble number of spekers, it my often be the cse tht mny of the spekers live in smll, reltively isolted communities or else form cliques within lrger communities with which they hve comprtively little interction. Deterministic models bsed on systems of differentil equtions, such s (11), might be unble to cpture the full rnge of possible behviours of the underlying system. In order to encpsulte vrition cused by such fctors, we dpt our model to investigte the stochstic nture of the dynmics of lnguge competition. To do so, we implement the system s n gent-bsed simultion model, n pproch tht hs found frequent ppliction in lnguge evolution studies (e.g. Hurford, 1989; Nowk, Plotkin & Krkuer, 1999; Wng, Ke & Minett, 2004). An gent-bsed simultion is model in which discrete elements, clled gents, represent selected entities or groups of entities of the underlying system tht is being modelled.
21 Unlike dynmicl systems such s (11), in which the globl behviour of the entire system is determined by single set of differentil equtions, the dynmics of gent-bsed models re described loclly in terms of how individul gents interct with ech of their neighbours. In the gent-bsed model of lnguge competition tht we introduce here, the gents correspond to the individul spekers tht comprise the popultion being modelled. Ech of the n gents dopts one of three possible sttes: monolingul in lnguge X (stte X), monolingul in lnguge Y (stte Y), or bilingul (stte Z). Wheres the deterministic model is described in terms of formule tht specify the rtes of trnsition in the proportions of individuls hving certin sttes, we design the gent-bsed model to use those sme formule to specify the probbilities with which ech gent mkes the trnsition to nother stte. For exmple, in the deterministic model, Z-bilinguls following the V-Model produce X-monolingul offspring t rte c ZX s X x (by eqution 7). In the gent-bsed model we re-interpret this to men tht Z-bilingul gents produce X-monolingul offspring with probbility c ZX s X x, where x now denotes the proportion of n gent s neighbours who re X-monolingul. Throughout the current section we will ssume tht the gents re ll neighbours of ech other, modelling popultion hving fully connected socil network. However, the re-interprettion of the trnsition rtes s probbilities llows us to model the impct of other socil structures on the dynmics of lnguge competition, n ide tht we pursue in Section 5. The trnsition probbilities of the V-Model re given by: ( X X) 1 Pr =, (14) ( Y Y) 1 Pr =, (14b) ( Z X) = c s x Pr, (14c) ZX ( Z Y) = c s y ZY X Pr, (14d) Y ( Z Y) = 1 c s x c s y Pr, (14e) ZX X ZY Y
22 where x, y nd z denote, respectively, the proportions of n gent s neighbours who re X-monolingul, Y-monolingul nd Z-bilingul. All other trnsitions, e.g., X Z nd X Y, hve probbility zero. The trnsition probbilities of the H-Model re: ( X X) = 1 c s y Pr, (15) XZ ( X Z) = c s y XZ Y Y Pr, (15b) ( Y Y) = 1 c s x Pr, (15c) YZ ( Y Z) = c s x YZ X X Pr, (15d) ( Z Z) 1 Pr =. (15e) All other trnsitions hve probbility zero. Agents undergo verticl trnsmission with probbility µ, the mortlity rte defined in Section 2.2; otherwise, they undergo horizontl trnsmission. The simultion is run s follows: The n gents tht comprise the popultion re ssigned initil sttes ccording to the selected initil proportions of spekers of ech type. We denote the initil proportions of monolingul spekers of X nd Y by x 0 nd y 0, respectively. Hving lso specified the vlues of the other prmeters, s X, c ZX, c ZY, c XZ, c YZ, nd µ, we set the simultion running itertively. At ech itertion, ech gent smples the sttes of ll its neighbours to determine its trnsition probbilities ccording to equtions (14) nd (15). Its stte is then rndomly updted ccordingly using the roulette wheel procedure (Goldberg, 1989). Once the simultion hs run for specified number of itertions, we identify the globl stte tht emerges: X-Monolingul (95% or more gents monolingul in X), Y-Monolingul (95% or more gents monolingul in Y), or Z-Bilingul ( mixture of both monolingul nd bilingul gents).
23 4.1 Dynmics of the Agent-bsed Model We now explin our pproch to estimting the probbility tht the system converges to ech stble stte by mens of n exmple. Figure 8 shows the evolution of the system during one run of the simultion for popultion of 1,000 gents of whom 35% re initilly X-monolingul nd 60% Y-monolingul, the reminder being bilingul. In this run, the system quickly converges to the stble equilibrium S* t x 0.30, y 0.45, z 0.25 bout which it then oscilltes. This represents n endngered lnguge X being mintined with bout 30% monolingul nd 25% bilingul spekers. Notice tht the trjectory, lso shown in the figure, follows the phse portrit of the deterministic system (11), on which it is superimposed, to lrge degree, indicting tht the behviour of the stochstic, gent-bsed model hs not diverged significntly from tht of the deterministic system. The sme initil conditions, however, sometimes led to the system converging to the stte Y* in which only lnguge Y hs ny spekers, s shown for second run in Figure 8b. Despite converging to different stble stte, the trjectory gin follows the direction field closely. This vrition in the finl stte of the system is due to the fct tht the initil stte (x 0 = 35%, y 0 = 60%) lies close to the boundry between the bsins of ttrction of the stble equilibri Y* nd S*. Slight perturbtions wy from the expected trjectory, indicted by the phse portrit, cn led to the system converging to stble equilibri tht differ from tht predicted by the deterministic system. By clculting the reltive frequency of convergence to ech stble equilibrium over mny runs of the simultion, the likelihood tht the system converges to ech equilibrium cn be estimted. Our primry concern is to determine not only wht intervention should be undertken but lso when intervention should be undertken in order tht n endngered lnguge be mintined. Consider Figure 9, which shows the impct on popultion of 1,000 gents, initilly consisting of 80% X-monolinguls nd 20% Y-monolinguls, of intervening to increse the sttus of X from s X = 0.2 to s X = 0.3. The figure indictes tht if the intervention is mde
24 fter the proportion of monolingul spekers of X hs fllen to 0.3 or lower (th X 0.3), then only lnguge Y cn be mintined. Mintennce of the two competing lnguges is possible if the intervention is mde on the intervl 0.3 < th X < 0.8, nd is inevitble only for intervention on the intervl 0.5 th X 0.6. Figure 9b, however, shows the effects of intervention whereby the sttus of X is enhnced from s X = 0.2 to s X = 0.4. In this cse, both lnguges cn be mintined with non-negligible probbility on the intervl 0.2 < th X < 0.8, the probbility rising to 1 on the intervl 0.4 th X 0.6. We observe tht incresing the post-intervention sttus of n endngered lnguge brodens the rnge of vlues of the intervention threshold, th X, for which both the endngered lnguge nd the lnguge with which it competes re both mintined. In prticulr, the lower bound on th X is decresed but the upper bound undergoes no significnt chnge. We observe the sme qulittive behviour for other sets of prmeter vlues. This suggests tht the more efficiently community cn bring bout n increse in the sttus of n endngered lnguge or, more generlly, implement ny of the strtegies proposed in Tble 1 the lter such intervention my tke plce in order tht mintennce be chieved. Even when such intervention is expected, ultimtely, to fil, the extinction of n endngered lnguge is not immedite. The rte t which lnguge looses spekers depends on the vlues of the system prmeters. Figure 10 illustrtes the impct of popultion size on the rte of extinction, plotting the frequency of occurrence of ech convergent stte s function of popultion size for one prticulr set of vlues of prmeters s X, c ZX, c ZY, c XZ, nd c YZ. For popultion of 100 individuls, lnguge X becomes extinct within 1,000 itertions with frequency ~35%; X is mintined long with the prestige lnguge with frequency 65%. As the popultion size is incresed, however, the frequency with which X is mintined grdully increses until, for n 3,500 individuls, the frequency of mintennce pproches 100%. The sme qulittive behvior is observed for other sets of prmeter vlues. We therefore conclude tht the probbility of mintennce over fixed durtion increses with popultion size.
25 5. The Impct of Socil Structure on the Probbility of Lnguge Mintennce Our simultions in the previous section were bsed on the ssumption tht ech gent hs complete knowledge of nd is influenced by the sttes of ll other spekers in the popultion undergoing lnguge competition. In effect, this is equivlent to ssuming tht the underlying socil structure is fully connected. Here, we investigte the impct of other selected socil structures on the probbility of lnguge mintennce. We represent socil structure s network whose nodes represent the individul spekers comprising the popultion. The edges of the network connect individuls who communicte with ech other. In prticulr, we model the socil structure s locl-world network (Li & Chen, 2003). Locl-world networks integrte into single prdigm both rndom networks (Erdős & Rényi, 1959) nd scle-free networks (Brbási & Albert, 1999), which, together with smll-world (Wtts & Strogtz, 1998) nd other network structures, re now finding ppliction in studies of socil nd sociolinguistic systems (e.g. Moody, 2001; de Bot & Stoessel, 2002; Smith, 2002; Ohtsuki et l., 2006; Cstelló et l., 2006). Locl-world networks re constructed recursively, dding nodes to the network one t time. As with scle-free networks, they re constructed using preferentil ttchment: when node is dded to the network, it is ssigned greter probbility of being connected to extnt nodes hving numerous connections thn to nodes hving few connections. This reflects n ssumption tht individuls prefer to interct with those spekers who themselves interct with mny other spekers. Unlike in scle-free networks, however, when node is dded to loclworld network it is connected preferentilly only to nodes within rndomly selected subset of ll extnt nodes, its locl-world. Thus individuls hve locl, rther thn globl, knowledge of the lnguge usge ptterns of other spekers in the popultion nd only interct with frction of the other spekers comprising the popultion. The procedure we use for constructing loclworld networks is described in Appendix A.
26 We now nlyze the effect of the initil proportions of monolingul spekers on the probbility of mintennce, strting with fully connected socil structures. Figure 11 shows the behviour of the system for popultion of 1,000 gents, with prmeter vlues set to = 1.0 nd s X = 0.4, with no intervention. The figure plots the estimted probbility of convergence to ech stte s function of the initil proportion of monolingul spekers of X; the remining spekers initilly ll spek Y. The figure clerly indictes tht stte in which both lnguges re mintined cn be chieved only with negligible probbility. Furthermore, for most initil proportions of spekers, x 0.7 or x 0.8, the system behves in the sme mnner s the deterministic system (11), with just one lnguge cquiring ll spekers with probbility 1. However, when the initil proportion of monolingul spekers of X lies in the rnge 0.7 x 0.8, there is grdul trnsition in the probbilities. This trnsition reflects the stochstic spect of the interctions mong the gents. As the popultion size increses, so the trnsition zone contrcts, nd the behviour of the system converges to tht of the deterministic system (11) for ll initil conditions. For loclly connected popultion, the behviour is indistinguishble from tht of the fully connected network shown in Figure 11. We observe the sme lck of impct of socil structure on the probbilities of convergence for other vlues of prmeters n, nd s 0. We therefore conclude tht, in the bsence of intervention, socil structure hs no significnt influence on which lnguge is mintined nd which lnguge dies. When the popultion intervenes to ttempt to mintin both competing lnguges, however, we find tht the underlying socil structure does ffect the behviour. Figure 12 shows exmples of the behviour for popultion of 1,000 gents tht intervenes to increse the sttus of the endngered lnguge X from 0.4 to 0.6 whenever the proportion of monolingul spekers of X flls below 0.5. Figure 12 highlights the behviour for fully connected popultion, clerly indicting the rnge of initil proportions of monolingul spekers of the endngered lnguge tht llow both lnguges to be mintined, 0.2 x 0 0.8, with mintennce being virtully
27 certin for 0.3 x 0 0.7. The sme qulittive behviour, in which single stble stte emerges with probbility ~1 over brod rnge of initil conditions, is observed for other vlues of prmeters n, nd s 0 for fully connected popultion. For loclly connected popultion, however, the behviour is less regulr. Figure 12b shows the grph for locl-world size of 50 gents, with the number of connections between ech incoming node nd its locl-world set to 20. We observe tht the rnge of vlues for which both lnguges cn possibly be mintined is the sme s for the fully connected popultion: 0.2 x 0 0.8. However, the pek probbility is somewht less thn 90% for x 0 = 0.3. The probbility of mintennce decys grdully for lrger initil proportions of monolingul spekers of the endngered lnguge to bout 50% for x 0 = 0.7. The probbility then decys rpidly to zero s x 0 pproches 0.8, s is the cse for the fully connected popultion. From this behviour we infer tht mintennce is more difficult to chieve within societies hving n underlying locl-world structure. Furthermore, the probbility of mintennce ppers to be mximl when intervention is undertken t the lst moment, but not so lte tht the opportunity is missed. Intervention is best implemented when the stte of the system is closest to the position of the stble equilibrium S* tht would be introduced by such enhncement; doing otherwise increses the probbility tht the system diverges from this equilibrium. Until the predictions of the model hve been fitted to empiricl dt, we hesitte to clim tht the probbilities quoted here indicte precisely the likelihood of lnguge mintennce being chieved within n ctul community. However, we do mintin tht comprison of the estimted probbilities of convergence for different sets of prmeter vlues nd initil conditions revel the qulittive behviour of the system, nd so inform us how better to intervene in order tht n endngered lnguge be mintined.