Demography and Language Competition

Transcription

1 Hman iology Volme 81 Isse 2 Special Isse on Demography and Cltral Macroevoltion rticle Demography and Langage Competition nne Kandler University College London a.kandler@cl.ac.k Follow this and additional works at: Recommended Citation Kandler nne (2009) "Demography and Langage Competition" Hman iology: Vol. 81: Iss. 2-3 rticle 5. vailable at:

2 Demography and Langage Competition bstract ttempts to describe langage competition and extinction in a mathematical way have enjoyed increased poplarity recently. In this paper I review recent modeling approaches and based on these findings propose a model of reaction-diffsion type. I analyze the dynamics of interactions of a poplation with two monolingal grops and a grop that is bilingal in these two langages. The reslts show that demographic factors sch as poplation growth or poplation dispersal play an important role in the competition dynamic. Frthermore I consider the impact of two strategies for langage maintenance: adjsting the stats of the endangered langage and adjsting the availability of monolingal and bilingal edcational resorces. Keywords langage competition langage extinction langage maintenance poplation growth poplation dispersal reaction dispersal competition model agent-based models. Cover Page Footnote I thank Stephan Shennan and James Steele the organizers of the rts and Hmanities Research Concil (HRC) Centre for the Evoltion of Cltral Diversity (CECD) workshop Demographic Processes and Cltral Change for the opportnity to present this work to an expert adience. I am also gratefl to James Steele for helpfl discssions and to two anonymos reviewers for constrctive comments on the manscript. Fnding was provided by a Leverhlme Trst Early Career Fellowship. This open access article is available in Hman iology:

3 Demography and Langage Competition nne Kandler 1 bstract ttempts to describe langage competition and extinction in a mathematical way have enjoyed increased poplarity recently. In this paper I review recent modeling approaches and based on these findings propose a model of reaction-diffsion type. I analyze the dynamics of interactions of a poplation with two monolingal grops and a grop that is bilingal in these two langages. The reslts show that demographic factors sch as poplation growth or poplation dispersal play an important role in the competition dynamic. Frthermore I consider the impact of two strategies for langage maintenance: adjsting the stats of the endangered langage and adjsting the availability of monolingal and bilingal edcational resorces. Langage competition and death is a phenomenon that can be observed worldwide. Lingists estimate that there are langages in the world today bt becase of an explosive spread of a few dominant langages (e.g. English or Chinese) at least half of them will become extinct in the 21st centry (Krass 1992). The processes that lead to the disappearance of langages have greatly accelerated over the past 200 years and this worrying rate of extinction is probably niqe to or time (Grenoble and Whaley 2006). nmber of different socioeconomic political and cltral factors can be identified as driving this decline of lingistic diversity. In the corse of globalization and of recent trends for rbanization and long-distance economic migration interactions between grops that speak different langages have increased and so has the need for a common langage of commnication. Some langages (e.g. English) have come to fill that role for historical economic and hegemonic reasons and as a conseqence have risen in importance in official and nonofficial matters. Their lexicons have conseqently expanded to represent all the paraphernalia of modernization frther enhancing their competitive advantage. In contrast minority langages are particlarly sbject to pressre and are at risk of extinction mainly becase speakers perceive an economic gain from shifting (Mfwene 2002). nmber of prominent lingists have called for an ecological approach to this global lingistic extinction crisis 1 HRC Centre for the Evoltion of Cltral Diversity Institte of rchaeology University College London London United Kingdom. Hman iology pril Jne 2009 v. 81 nos. 2 3 pp Copyright 2009 Wayne State University Press Detroit Michigan key words: langage competition langage extinction langage maintenance poplation growth poplation dispersal reactiondispersal competition model agent-based models.

4 182 / kandler (however they differ in what they mean by that term) and for the development of a therapetic nderstanding for the process of extinction (Fishman 2001). Langage extinction may be cased by the death of the poplation speaking the langage or by langage shift (Tsnoda 2004). In this paper I focs on langage shift which is defined as the process by which members of a commnity in which more than one langage is spoken abandon their original langage in favor of another (Tsnoda 2004). The reasons for langage shift are complex and Fishman (1964: 49) has stated that it is crrently impossible to specify in advance an invariant list of psychological social and cltral processes or variables that might be of niversal importance for an nderstanding of langage maintenance or langage shift. ccording to Crawford (1996) there seems to be no established and comprehensive theory of langage shift especially in terms of cases and varying conditions that might prevent them. brams and Strogatz (2003) addressed the problem of langage shift and extinction from a different perspective by characterizing and modeling the dynamics of langage competition in a mathematical way. They presented a two-langage competition model to explain historical data on the decline of endangered langages. The mathematical simplicity and despite some obvios nrealistic assmptions convincing fits of their model to empirical data have generated a brst of attempts to model the dynamics of langage competition. I start with an overview of recent mathematical modeling approaches in the field of langage competition (I try to se as far as possible a common notation throghot the description of the different models to make them comparable). However I do not claim that this is a complete overview. The reviewed models still lack lingistic reality and are often criticized by lingists. I propose and analyze a reaction-dispersal competition system. Nevertheless the proposed model does not solve all the problems and I address some critiqes. I analyze the reslting competition dynamic and am interested in certain characteristics sch as extinction time. In addition I demonstrate the effects of demographic factors sch as initial abndance poplation growth and dispersal on the langage competition and investigate the crcial aspect of langage maintenance. Recent Modeling pproaches brams and Strogatz (2003) generated a brst of modeling attempts into the dynamic of langage competition. However in 1990 aggs and Freedman had already pblished a (mainly overlooked) model based on the predator-prey paradigm for describing a sitation in which a bilingal poplation grop interacts with a monolingal poplation grop. The main idea is that the dynamics of the growth of the bilingal and monolingal poplations is determined by bonded birth-death processes with separate carrying capacities for each poplation grop a conversion mechanism and emigration processes. aggs and Freedman (1990) proposed the following model:

5 Demography and Langage Competition / 183 n t D H n L n nn 2 C n C t D H n L n 1 n 2 C C C C C C C nn 1 n Pn C C Pn C C (1a) (1b) where n is the freqency of monolingal individals speaking langage and n C is the freqency of bilingal individals in the poplation. The coefficients i D i and H i describe the birth death and emigration rates of each grop respectively. The parameter P defines the rate at which children of bilingal parents are raised monolingal [denoted as the infant langage acqisition parameter by Wybrn and Hayward (2008)] and is the rate at which monolingal individals convert to being bilingal that is the rate at which they learn the second langage [denoted as the noninfant langage acqisition parameter by Wybrn and Hayward (2008)]. The model in Eqs. (1) sggests that coexistence between the poplation grops is possible if the conversion rate and the rate at which children of bilingal parents are raised monolingal P are both moderate. Only extreme parameter vales lead to the extinction of one of the langage grops. Frthermore aggs and Freedman (1990) fond that small emigration rates do not alter the competition dynamics; however if either grop experiences large emigration then it is possible that this poplation grop will go extinct. aggs and Freedman (1993) generalized their model and considered the dynamics of the interactions of a poplation with two monolingal grops (in which one langage is assmed to be the high-stats langage) and a grop that is bilingal in these two langages in a closed environment. They conclded that different environmental conditions favor different dynamics. They fond that besides the extinction of the monolingal grop speaking the low-stats langage and of the bilingal grop coexistence between all three poplation grops and coexistence between the monolingal grop speaking the high-stats langage and the bilingal grop were possible. El-Owaidy and Ismail (2002) generalized this model by incorporating a third monolingal poplation grop. Recently Wybrn and Hayward (2008) revisited aggs and Freedman s (1993) model. They analyzed the otcomes of the aggs-freedman model for varying conversion rates and defined for different scenarios (bilingal death bilingal prestige bilingal maintenance bilingal shift) in the long-term ftre of the bilingal poplation. Frthermore they applied the aggs-freedman model to varios real-life sitations (e.g. the English and Welsh-English competition in Wales). fter estimating the model parameter from external sorces Wybrn and Hayward (2008) predicted the state of the bilingal poplation grop that is which of the for long-term scenarios wold be most likely. In most cases the prediction coincided with the observed sitation. In addition Wybrn and Hoyward (2009) applied a model based on operational research methodology to the problem of langage competition and langage planning.

6 184 / kandler brams and Strogatz (2003) developed a simple deterministic model for describing the changes in the patterns of langage sage within a poplation in which two langages compete. Their aim was to explain historical data on the decline of endangered langages and to qantify indicators of langage endangerment so that sefl langage-preservation programs cold be set p at an early stage. They defined langage competition as competition for speakers; langage itself was assmed to be fixed. The dynamics of the competition is described by the temporal change of the proportion of speakers of both langages which reslts in dn np ( n s ) np ( n s ). (2) dt The terms n and n stand for the freqencies of speakers of langage and respectively and it is assmed that n n 1. This condition implies a fixed poplation size. The term P models the rate at which an individal shifts from langage to langage per time nit. brams and Strogatz (2003) assmed that this shift rate is determined by the attractiveness of langage the target of shifting. They defined attractiveness by the proportion of speakers of langage and its social stats s and described the shift rate P with the power law a P ( n s ) cn s. (3) The stats parameter s reflects the social or economic opportnities afforded to the speakers of langage relative to langage (brams and Strogatz 2003). The parameter s is defined in the same way and it yields s s 1. Ths the higher the proportion of speakers of langage and the higher its social stats the higher its attractiveness and therefore the higher the likelihood that speakers of langage will shift to langage. The exponent a models how the attractiveness of langage scales with the proportion of speakers of langage and the coefficient c indicates the peak rate at which speakers of langage shift to langage. The opposed shift rate P is defined analogosly by a P cn s. (4) The analysis of dynamic systems sch as model (2) encompasses the determination of the stable eqilibria expressing the freqencies which do not change over time anymore and are robst against small pertrbations. The brams- Strogatz model predicts that one langage (depending on the social stats parameters s and s and on the distribtion of the initial freqencies) will always acqire all speakers in the poplation casing the langage with which it competes to become extinct. To evalate the significance of their approach brams and Strogatz (2003) fitted the model to diachronic data collected for Scottish Gaelic Welsh and Qecha and fond a good coincidence between the predicted and observed declines of the langage sage. Last they sggested that langage maintenance can be achieved by controlling for the social stats of the endangered langage. On the basis of these reslts the brams-strogatz model appears to be an appropriate approach for modeling the dynamics of langage competition. However it incldes the following idealistic assmptions which may limit the ex-

7 Demography and Langage Competition / 185 planatory vale of the model: (1) Langages are assmed to be fixed; (2) the poplation is highly connected with no spatial or social strctre; (3) all speakers are monolingal; (4) the poplation size is assmed to be constant; and (5) there is no distinction between different ses of langage in different social contexts. Sbseqent modeling approaches generalized the brams-strogatz model by addressing one or more of these shortcomings. These approaches can be categorized into two grops: those concerned with a more realistic modeling of the demography of the poplation and those concerned with a more realistic modeling of the process of langage shift. The important role of demography in langage competition can be powerflly demonstrated with the example of the farming-langage hypothesis (cf. e.g. Diamond and ellwood 2003; Renfrew 1987). In the period known as the Neolithic revoltion agricltre qickly spread to cover large parts of the world. rchaeological and lingistic evidence (ellwood and Renfrew 2003) sggests that the dispersal of farming was accompanied by the dispersal of the Indo-Eropean langage. However it shold be mentioned that this hypothesis is not withot controversy [for a discssion on this topic see for example Shose (2001)]. s soon as the advantageos technology farming was established in its centers in the Near East sia and frica it spread ot in a traveling wavelike pattern. Whether the concept of farming was an advantageos technology in the Neolithic is still sbject to debate becase ntritional standards of Neolithic poplations were generally inferior to those of hnter-gatherers and life expectancy may have been shorter. Spported by a mch higher fertility and an ability to sstain higher poplation densities farmers replaced hnter-gatherers. s farming spread ot from its centers langage was pshed ot with it and the Neolithic became the first period of lingistic homogenization (Nettle and Romaine 2000). Frthermore farming and conseqently the spread of langage were stopped only where the environment was not sitable or where geographic bondaries made it impossible. This example shows how the spread and therefore competition of langages can be determined by the demographic characteristics of the competing sbpoplations. In this context ckland et al. (2007) developed a model of cltral hitchhiking that can serve as a nll model for explaining the spread of the Indo-Eropean langage. Patriarca and Leppänen (2004) acconted for the fact that the spread of a langage is inflenced by the dispersal of its speakers and they generalized the brams-strogatz model by introdcing spatial dependence. This reslts in a reaction-diffsion system of the form (5a) d P( n s) P( n s) t (5b) d P( n s) P( n s) t where and are the space- and time-dependent freqencies of both langages and P and P describe the shift rates (which are still determined by the overall

8 186 / kandler freqencies n and n respectively) as in the brams-strogatz model. The diffsion components d and d ( is the Laplace operator) model the spatial dispersal of the speakers based on the random walk hypothesis. The overall poplation size is assmed to be constant. The analysis of the model in Eqs. (5) shows that again only the extinction states are stable eqilibria. Frthermore Patriarca and Leppänen (2004) fond that if the considered domain is divided into two distinct zones and if the shift rates P and P differ from zone to zone depending on the local poplation freqency then two langages with different social statses can coexist if langages acqire speakers in different locations. Pinasco and Romanelli (2006) developed an ecological model of the Lotka- Volterra type to explain cases in which both langages srvive in only one zone of competition: n (6a) n rn 1 cnn t K n (6b) n rn 1 cnn. t K gain n and n are the time-dependent freqencies of both langages. This model introdces an additional logistic growth term whereby new speakers of each langage are recrited not jst by shifting bt also by reprodction. To maintain finite poplation sizes reprodction is modeled as a logistic process with the carrying capacities K and K respectively. The assmed langage shift dynamic is similar to the brams-strogatz model. Pinasco and Romanelli (2006) fond a stable eqilibrim in which both langages coexist. Kandler and Steele (2008) incorporated both demographic factors (poplation growth and spatial dispersal) into a reaction-diffsion system and qestioned the assmption of the Pinasco-Romanelli approach that there are two separate carrying capacities K and K for speakers of langages and. They assme only one carrying capacity K to describe the maximm poplation size that can be spported by a given environment withot redcing its ability to spport the same poplation in the ftre (Ehrlich 1994) regardless of the langage spoken. That means that the time- and space-dependent freqencies and of both langages have to flfill the condition K (7) where K is for the common carrying capacity. This consideration leads to a reaction -diffsion system of the form d r1 c t K d r1 c. t K (8a) (8b)

9 Demography and Langage Competition / 187 nalogos to the Patriarca-Leppänen approach the diffsion terms d i i model the spatial dispersal whereas the terms r i i {1 [ i /(K j )]} model the intrinsic growth behavior of each sbpoplation. The competition coefficient c 0 reflects the social stats differences of both langages. The analysis of the model in Eqs. (8) shows that coexistence as in the Pinasco-Romanelli approach is no longer possible. Langage cannot resist the permanent conversion of speakers to langage and the extinction of the lower stats langage is predicted. However the extinction times and the temporal and spatial corse of the extinction process depend crcially on the demographic parameters d d r r and c. Frthermore in adapting Patriarca and Leppänen s reslts Kandler and Steele (2008) showed that spatial heterogeneity (in the form of spatially varying competition behavior) can affect the interaction dynamic. Kandler and Steele fond that coexistence is possible in sitations in which the attractiveness or dominance of langages changes between regions. The reasons for dominance can relate to political social and/or economic factors. n example is the sitation in the asqe Contry which enjoys sbstantial cltral and political atonomy and where the asqe langage is protected by laws and by the reglatory activity of an academy set p to maintain it. The asqe langage is an isolate nrelated to the Indo-Eropean langages also spoken in this and adjacent regions. It is plasible that this makes it more difficlt for speakers in one grop to learn the other grop s langage and that this has also impeded rates of shifting [althogh Kandler and Steele (2008) did not explicitly consider sch factors]. Recently Patriarca and Heinsal (2009) pblished a generalization of the Patriarca-Leppänen model by adding a logistic growth term (as in the other models described here) and an advection term. Their aim was to examine the inflence of geography on langage competition based on hman dispersal models. They showed that factors that are not related to the cltral transmission process sch as initial poplation distribtion or geographic bondaries and inhomogeneities can strongly affect the dynamics of langage competition. They obtained sitations of coexistence (assming a fixed poplation size) in which a geographic bondary (e.g. a montain chain) divides the area into two regions and for historical reasons the sbpoplations speaking langages and are initially localized on opposite sides of the barrier. We have seen that the incorporation of demographic aspects of langage competition can change the reslt of the original brams-strogatz model significantly. cconting for spatial poplation dispersal even in its simplest mathematical description as a diffsion process can reverse the competition reslt whereas spatial heterogeneity (whether in the form of heterogeneos growth behavior or spatially varying attractiveness of langages) can lead to coexistence. Nevertheless the reviewed models fail to address crcial social and lingistic aspects. Wang and Minett (2005) made the criticism that these dynamic models do not accont for bilingalism sociolingistic factors sch as social strctre mltiple registers of speech and the effect of heterogeneos strategies both at the level of individals and at the level of policymakers. Following this critiqe the second grop of

10 188 / kandler generalizations of the brams-strogatz model dealt with a more realistic modeling of the process of langage shift in particlar the incorporation of bilingalism. Mira and Paredes (2005) sggested a model in which the two langages that compete for speakers are partly mtally intelligible. When two langages possess a certain degree of similarity monolingal speakers of one langage can sometimes commnicate effectively with monolingal speakers of the competing langage which allows monolingal speakers to become bilingal. Mira and Paredes assmed that apart from the proportion of speakers of the competing langage the rate at which monolingal speakers shift to the bilingal sbpoplation depends on the degree of similarity between the langages. The model shows that for a sfficiently large degree of similarity a stable eqilibrim is obtained in which both bilingal speakers and monolingal speakers of the more prestigios langage srvive. Mira and Paredes fitted their model to diachronic data collected for Castilian Spanish and Galician and fond a good coincidence. Minett and Wang (2008) qestioned the applicability of the brams-strogatz model to general sitations of langage competition becase competing langages are often mtally nintelligible. Minett and Wang extended the brams-strogatz model by inclding bilingalism explicitly. This approach encompasses three sbpoplations two monolingal grops (denoted and ) and one bilingal grop (denoted C). Minett and Wang assmed that langage is transmitted vertically or horizontally. To determine which individals followed vertical transmission and which followed horizontal transmission they introdced a mortality rate at which adlts are replaced by children. These assmptions led to the following system of differential eqations: dn c C ( 1 n n ) P ( n s ) ( 1 ) c C n P ( n s ) (9a) dt dn c C ( 1 n n ) P ( n s ) ( 1 ) c C n P ( n s ). (9b) dt The terms n and n describe the time-dependent freqencies of langages and respectively and n n n C 1 where n C is the freqency of the bilingal poplation. The shift rates P and P are the same as in the brams-strogatz model. This means that in this model too the attractiveness of a langage determines the strength of vertical and horizontal transmission. The model in Eqs. (9) predicts the extinction of one langage regardless of the initial conditions. ilingalism is not able to prodce coexistence. Inspired by the reslt of brams and Strogatz (2003) that an endangered langage can be preserved by controlling for its social stats Minett and Wang (2008) trned to the important aspect of langage maintenance. They fond that increasing the social stats of the endangered langage and modifying the parameters c C c C c C and c C (which can be associated with different intervention strategies) can lead to stable coexistence of the two monolingal sbpoplations. Smmarizing these generalizations of the brams-strogatz model point ot clearly that both demographic factors and lingistic and social aspects can play an

11 Demography and Langage Competition / 189 important role in explaining langage competition and death. Neglecting one of them can change the competition dynamics drastically. Nevertheless in sitations in which every individal experiences the same homogeneos (cltral) environment and is exposed to both langages at some point most of the models indicate that one langage will go extinct. t the process of extinction and especially the extinction time depend crcially on the chosen model setp. Coexistence is achieved nder consideration of poplation or environmental heterogeneity or external interventions. Having these facts in mind I believe that demographic aspects and realistic lingistic and social transmission processes shold not be considered separately. Therefore in the next section I propose a reaction-dispersal competition model that incorporates bilingalism vertical and horizontal langage transmission [as sggested by Minett and Wang (2008)] social strctre and demographic factors sch as poplation dispersal and poplation growth. esides these dynamic models which describe the competition dynamics deterministically sing copled differential eqations another grop of powerfl approaches based on agent-based simlations has been developed. Whereas the previosly reviewed approaches act on the poplation level and deal with the expected general pattern of the langage competition agent-based simlations act on the individal level and model the actions and interactions of agents in a network with a view to assessing their effects on the system as a whole. gent-based models focs on a realistic modeling of the contact sitation to find appropriate social strctres throgh modeling the nderlying interaction network incorporating bilingalism and modeling langage evoltion. Mch work on agent-based modeling of langage competition was done by Staffer and Schlze [for a review of their models see Schlze et al. (2008) and Staffer and Schlze (2005)]. They sed a bit-string approach in which langages are described by strings of F bits. ll their models exhibit the following common mechanisms (cf. Schlze et al. 2008). langage is characterized by F independent featres (which are identified with an independent grammatical element) where each featre can take one of Q different vales. In each iteration each of the F featres is changed with the probability p. With probability q this change is deterministic and the vale of the featre is simply transferred from another individal in the poplation; on the contrary with probability (1 q) the change is chosen randomly. Langage shift is determined by the density - dependent probability (1 x) 2 r where x is the proportion of the poplation speaking the individal s native langage and r is the shift rate. Staffer Schlze and colleages analyzed this basic model nder different assmptions for the poplation strctre geography migration behavior or poplation growth bt were mainly interested in the competition of many langages and the distribtion of the world s langages. One of the findings was that the simlated distribtion of the nmber of langages spoken by s individals cold be described by a parabola in a log-log plot and matched empirical estimates. In the context of the langage size distribtion de Oliveira et al. (2006) developed a simlation model based on the idea that the fitness of a langage is

12 190 / kandler proportional to the nmber of its speakers and that the mtation probability is inversely proportional to the langage s fitness. They fond that the nmber of langages spoken in an area varied as 0.4. This reslt coincided with empirical data. The competition of many langages was also stdied nmerically by Nettle (1999). He hypothesized that the temporal evoltion of the nmber of langage grops L can be described by the eqation dl/dt 70/t L/20. However Staffer and Schlze (2005) applied their model to the sitation stdied by Patriarca and Leppänen (2004). Initially half of the domain is occpied by langage and half of the domain is occpied by the opposite langage. There is no stats difference between the two langages bt Staffer and Schlze introdced a local conformist bias to the competition dynamic. They obtained coexistence of the two dominant langages and in which both langages were spatially separated and interacted only in a small transition zone. The initial separation of both langages is a crcial assmption for that reslt. Schlze and Staffer (2006) stdied the srvival of a minority langage that has no stats disadvantage. They conclded that coexistence between a majority langage and a minority langage can be achieved if speakers of the minority langage refse to shift their langage as soon as the freqency of the majority langage exceeds a certain threshold. Kosmidis et al. (2005) developed a similar agent-based approach for the competition of two langages. Every agent has the capacity to speak two langages. Here langages are characterized by a vocablary of 10 words and each time an agent interacts with another the agent can learn a word from the other. It is assmed that the agent s fitness (defined by the agent s reprodctive sccess) increases by learning words from the other langage. This model leads mainly to coexistence sitations. Under the assmption of a finite poplation size with no birth and death the poplation will speak on average five words of each langage. The inclsion of birth and death processes reslts in a sitation in which nearly everyone is bilingal (that means everyone speaks all 10 words of each langage). Kosmidis and co-workers fond that one langage goes extinct if there are disadvantages in the initial freqency and fitness level of its speakers or if demographic stochasticity is added. Schwämmle (2005) also sed a bit-string approach bt was interested in the effect that biological aging has on langage competition. ging was incorporated throgh the Penna model (Penna 1995) and the model approved the fact that langages are learned more easily in yoth than in old age. This approach is seen as a bridge between the langage learning and langage competition literatre (Schlze et al. 2008). In direct relation to the basic brams-strogatz model Staffer et al. (2007) developed a microscopic version and analyzed this system on a flly connected network and on a d-dimensional lattice. flly connected network is a system in which all nodes are connected to each other. That means that agents can interact directly with all others. On the contrary in a d-dimensional lattice only the 2d nearest neighbors are connected and ths agents interact only locally. Staffer

13 Demography and Langage Competition / 191 and co-workers assmed that in each iteration one agent is chosen at random and will change its langage according to the freqency-dependent probabilities P and P defined by the brams-strogatz approach. Note that nder the assmption of a d-dimensional lattice P and P are determined by sing local freqencies. Staffer et al. (2007) showed that this microscopic version coincides with the original brams-strogatz model for sitations in which both langages possess different social statses (s s 0.5). If the competing langages have the same stats then the differential eqation formlation reslts in the nstable coexistence state ( ) whereas the agent-based model leads to the more realistic sitation in which langage acqires all speakers in half of the simlations and langage acqires all speakers in the other half of the simlations. The Staffer et al. (2007) simlation model was extended by Castelló et al. (2007) by introdcing bilingalism and social strctre. Castelló s agents were allowed to speak either one of the langages or or to speak both and besides d-dimensional networks there were small-world networks which acconted for short- and long-range interactions. In small-world networks most nodes are not neighbors of one another bt most nodes can be reached from every other node by a small nmber of steps (cf. e.g. Watts and Strogatz 1998). In each iteration one agent i was chosen at random and the local freqencies n (i) and n (i) of both langages arond that agent were determined. Then the agent changed its state ( denotes the state of speaking langage of speaking langage and C of being bilingal) according to the freqency-dependent probabilities P C i 05. n ( i ) P C i 05. n ( i ) P C i 051. [ n ( i )] P C i 051. [ n ( i )]. (10a) (10b) (10c) (10d) Note that an agent can only change from being monolingal to being bilingal and vice versa. The shift rates are symmetric which implies that both langages possess the same social stats. The analysis of the Castelló et al. (2007) model shows that neither bilingalism nor social strctre is able to prodce coexistence between the two eqalstats langages. Frther Castelló and co-workers conclded that bilingalism is not a stable strategy. ilingal agents place themselves at the bondaries between the monolingal spatial domains to favor commnication between them. Social strctre in the form of small-world networks acts as a way of accelerating the extinction process. Minett and Wang (2008) developed a microscopic analoge to their differential eqation approach. Their aim was to extend the reslts from the continos approach by analyzing the range of behaviors of langage competition that can reslt from specific intervention mechanisms and initial conditions. Similar to the Castelló et al. (2007) approach Minett and Wang assmed that each agent had the

14 192 / kandler possibility of being monolingal in langage or or of being bilingal. gents nderwent vertical transmission with probability ; otherwise they nderwent horizontal transmission. In each iteration each agent sampled its neighborhood to determine the shift probabilities according to a P c s n C C a P c s n C C a P c s n C C a P c s n. C C (11a) (11b) (11c) (11d) The state of the agent was then changed randomly. Minett and Wang (2008) analyzed the model on a flly connected network and on a local-world network to accont for social strctre. In contrast to small-world networks a local-world network is an evolving network with local rather than global connections. It reflects the assmption that individals have local rather than global knowledge of the langage sage patterns of other speakers in the poplation and only interact with a fraction of the other speakers making p the poplation (Minett and Wang 2008). Minett and Wang fond that the more efficiently a commnity is able to increase the social stats of the endangered langage the later sch interventions have to take place. Frther they conclded that in a system withot intervention the social strctre does not inflence the competition dynamic. However interventions are more likely to be sccessfl in a poplation whose social strctre can be described by a flly connected network. Most of the mathematical approaches sed to describe langage competition are based on either differential eqations or agent-based simlations. However the focs of both grops is different. s already mentioned the continos systems of differential eqations describe the general expected behavior of the competition on the poplation level whereas agent-based simlations model the interactions between agents and their inflences on the whole system on an individal level. Differential-eqation-based approaches are often criticized for their deterministic natre negligence of a finite poplation size and inability to captre all possible behaviors occrring in langage competition. Minett and Wang (2008) stated that if maintenance of endangered langages is considered speakers might live in small relatively isolated commnities or form cliqes within larger commnities with which they have comparatively little interaction and it might not be possible to model these effects with systems based on differential eqations. That is ndobtedly tre for basic models (as it wold be for basic agent-based simlations) bt it has already been shown in other fields that with appropriate adjstments many of these shortcomings can be addressed. (The effects of different poplation sizes cannot be modeled with systems of differential eqations.) However the effects of small poplation sizes cannot be modeled with systems of differential eqations. So there is no right model to describe langage competition. oth grops have proved to have powerfl tools (with their

15 Demography and Langage Competition / 193 specific shortcomings) and we shold see them not as competing bt as complementary approaches that can benefit from the respectively gained insights as was demonstrated powerflly by Minett and Wang (2008). Reaction-Diffsion Competition pproach The Model. In this section I model the dynamic of langage competition in a temporally and spatially homogeneos environment sing a reaction-diffsion competition model. The se of this model type was encoraged by earlier stdies of langage competition (e.g. Kandler and Steele 2008; Patriarca and Heinsal 2009; Pinasco and Romanelli 2006) cltral hitchhiking (ckland et al. 2007) or prestige bias (Ihara 2008) that exploit similar methods. Reaction-diffsion competition models inclde growth dispersal and competition components which collectively are well sited to captring aspects of the spread and competition of langages in a poplation. mathematical detailed description of these approaches can be fond in for example Freedman (1980) and Mrray (1996). I propose the following model: d r 1 c ccc t K d r 1 c ccc t K C C (12a) (12b) (12c) C C dcc rcc 1 ( cc cc ) C ( c c ) t K with the bondary conditions i /n 0 and x D. The expression /n describes the oter normal derivation. These bondary conditions model sitations in which no diffsion beyond the bondary D of the domain D is possible. The time- and space-dependent variables and stand for the freqencies of speakers of langages and respectively and C describes the freqency of bilingal speakers. The terms i /t indicate the temporal change of the freqencies. In addition the spatial dispersal behavior of the sbpoplations is described by the diffsion components d i i ( denotes the Laplace operator). That means that spatial dispersal has only a local dimension. I analyze the effects of nonlocal dispersal behavior by replacing the diffsion components with an integral formlation that allows for short and long-range dispersal. The reaction terms 1 r K C c c C C 1 r c c C C K C (13a) (13b)

16 194 / kandler Figre 1. Schema of the assmed shift mechanisms in the proposed model [Eqs. (12)]. 1 C r (13c) C C ( c C c C ) C ( c c ) K describe the growth behavior of the three sbpoplations for which two main factors are involved. The first factor in Eqs. (13) is an intrinsic growth component that models copled biological and cltral reprodction within each sbpoplation. This component has the form r 1 i (14) i i Kj k which denotes a bonded logistic growth process with an intrinsic natral rate of increase r i defined as the difference between birth and death rates. The growth of the whole poplation (i.e. of C ) is restricted to the common carrying capacity K which defines an pper limit of the poplation size (cf. Kandler and Steele 2008). The second factor in Eqs. (13) captres the dynamic of langage shift and is illstrated in Figre 1. I assme that langage shift mst involve a bilingal transition state and that the loss of individals of one sbpoplation is the gain of another sbpoplation. The solid arrows indicate fndamental shift rates from the two monolingal sbpoplations to the bilingal stage and the dashed arrows represent the rate of loss of bilingal speakers to each of the monolingal sbpoplations. In more detail langage shift is described by the density-dependent conversion terms c ij i j. The coefficients c and c describe the rate at which monolingal individals become bilingal. In total a proportion of (c c ) of the monolingal poplation shifts to the bilingal poplation. On the contrary the bilingal sbpoplation loses a proportion of c C C c C C to

17 Demography and Langage Competition / 195 the monolingal poplation. Following brams and Strogatz (2003) the rates c C c C c and c reflect the prestige or socioeconomic advantage that accre to speakers of langage and langage respectively. Frther I introdce two variables s and s that qantify the stats or prestige differences between the two langages. For simplification I set s s 1 (brams and Strogatz 2003) and sbstitte s with 1 s. I assme c c ( 1 s ) C C c c s C C c c s c c ( 1 s ). (15a) (15b) (15c) (15d) The higher the social stats of a langage the lower the loss rate to the bilingal sbpoplation and the higher the gain rate from the bilingal grop. Following Minett and Wang (2008) I introdce coefficients c C c C c and c which allow s to model the peak attractiveness of a langage. I define the overall balance of competitive advantage to speaking each langage on the basis of these conversion rates: For example langage can be assmed to be more advantageos if it holds that c c and c C c C. This implies that monolingals who speak langage are less likely to become bilingal and that bilingals are more likely to shift to langage. In more detail langage transmission occrs vertically or horizontally and I explain in the following discssion how this is incorporated into the model in Eqs. (12). In general vertical transmission describes the passing of cltral traits from generation to generation. We assme that children of monolingal parents will be raised monolingal. In these cases biological and cltral reprodctions coincide. However bilingal parents may choose to raise their children monolingal or bilingal. Vertical transmission is modeled in Eqs. (12) sing the logistic growth processes and the freqency-dependent conversion terms c C C and c C C. The loss of bilingal offspring to langage occrs at a rate c C and the loss to langage occrs at a rate c C C. Horizontal transmission (the spread of cltral traits between individals of the same generation) is cased by the need for a common commnication base when speakers of different langages are present in the same domain. Then monolingal speakers are encoraged to learn the other langage and therefore to become bilingal. Horizontal transmission is incorporated into Eqs. (12) by the freqency-dependent conversion terms c and c. s for vertical transmission the rate at which speakers choose to learn the second langage depends crcially on the sbpoplation s attitde toward learning a foreign langage and again the attractiveness of the langages. The effect of horizontal transmission is to swell the bilingal sbpoplation as monolingals learn a second langage. I do not accont for extended diglossia therefore assming that individals se the langages in all social contexts eqally.

18 196 / kandler Reslts In the following discssion I analyze the competition dynamic that is modeled by Eqs. (12) for sitations in which one langage has a stats advantage. I assme a spatially and temporally homogeneos environment which is reflected by constant parameters in the considered model. To carry ot the analysis I implemented the model in Eqs. (12) in C and solved it nmerically sing finite element methods. The only stable eqilibrim states that are obtained are the extinction states (K 0 0) and (0 0 K). The proposed model predicts that depending on the attractiveness of both langages the demographic parameters of the sbpoplations and their initial distribtions one langage will acqire all speakers over time. Interestingly that does not have to be the high-stats langage. For the sake of illstration Figre 2 shows an example for the temporal and spatial competition dynamic. ssme that the parameter constellation is K 1 r 0.06 r 0.03 r C 0.05 c C 0.05 c C 0.02 c 0.03 and c 0.05 which models a sitation in which speakers of langage have an intrinsic advantage. Frthermore assme that initially langage is the langage with the most speakers in the considered domain D bt that the high-stats langage has entered the poplation in a small region. Figre 2 shows the spatial distribtion of the sbpoplation freqencies for d d d C 10 4 (straight lines) and for d 10 2 d d C 10 5 (dashed lines) at two different time points. For the sake of simplification and better illstration Figres 2 6 and 7 show cts throgh the two-dimensional rectanglar domain D [0 1] [0 1] at x When all sbpoplations show the same dispersal behavior it is obvios that becase of its competitive advantage the sbpoplation speaking langage grows in the center of the domain and local diffsion cases a steady expansion of the zone in which langage was fond at high enogh freqencies for it to prevail. With time the contact and mixing zone is shifted toward the edges of the domain with extinction of langage and then of bilingalism as the long-term otcome. However for d 10 2 and d d C 10 5 (dashed lines in Figre 2) the reslt of the competition is reversed. In this sitation the relatively greater diffsivity (d 10 2 ) of the speakers of langage cases dramatic diltion of the initial concentration of its speakers in the center of the domain. The intrinsic growth rate r is not able to compensate for this and the density-dependent dynamics predominate leading to the extinction of langage. These findings raise the qestion of nder which conditions the initially mainly spoken bt low-stats langage is able to resist the presence of the high-stats langage. I explore this qestion by analyzing the competition otcome for different parameter vales for d (a measre of the mobility of speakers of langage ) s (the social stats of langage ) and its initial distribtion. ssme again that langage has entered the poplation in a small area and that the freqency at which langage is present in this area initially is varied. Figre 3 shows the interface that separates the area of attraction of the extinction

19 Demography and Langage Competition / 197 Figre 2. Spatial freqency distribtions of the three sbpoplations at different times. Solid lines represent the competition behavior for d d d C 10 4 and dashed lines describe the behavior for d 10 2 d d C t the beginning the high-stats langage is present only in a small area (D [ ]) and at a low freqency ( 0.3 for x D). The social stats variable s is assmed to be 0.3. states (K 0 0) (above the srface) and (0 0 K) (below the srface). We observe a nonlinear relationship between the three parameters. larger mobility of the sbpoplation of speakers of langage can be balanced by a higher social stats and/or a higher initial concentration. Smmarizing besides the social stats difference the dispersal behavior and the initial distribtion of the sbpoplations

20 198 / kandler Initial freqency of langage d 0.6 x 10 4 Social stats s Figre 3. Srface that separates the areas of attraction of the extinction states (K 0 0) (area above the srface) and (0 0 K) (area below the srface). play an important role in the competition behavior. If the low-stats langage is sfficiently established in the poplation it can prevail in competition with a more intrinsically advantageos. We are able to predict which langage will acqire all speakers in the long rn. However the time ntil the extinction state is reached may vary greatly. Figre 4 shows the extinction times for different vales of the parameters s and d and an initial freqency of speakers of langage of 0.1 (Figre 4a) and 0.4 (Figre 4b). It is obvios that the peak extinction time is reached for the parameter constellation where langage otcompetes langage for the first time. This effect is cased by the chosen initial distribtion. Langage has to conqer a larger area than langage becase it is present only in a small area at the beginning. Then a frther decrease of s (the relative social stats of langage ) accelerates the extinction process. In contrast when the low-stats bt initially more abndant langage wins the competition we observe an increase in the extinction time if langage s stats is increased. comparison of Figres 4a and 4b shows the inflence of the initial condition. The higher the initial abndance of langage the lower the social stats s or the dispersal rate d has to be to otcompete langage. Role of ilingalism. Langage shift is said to presppose the existence of a transitional stage of bilingalism involving receding langage and the replacing

21 Demography and Langage Competition / 199 Extinction time x d Social stats s Figre 4. Extinction times depending on the social stats and the diffsivity d of the sbpoplation speaking langage for an initial freqency of (a) 0.1 and (b) 0.4. The light gray circles indicate the sitations in which langage will acqire all speakers over time whereas the dark gray circles indicate sitations in which the low-stats langage will acqire all speakers. one (e.g. Campbell 1994). In this context Tsnoda (2004) asked whether it is possible for langage shift to happen withot a transitional stage of bilingalism. It appears that there is no incidence reported (Tsnoda 2004). It is often stated that it takes three generations for langage shift to be completed (renzinger 1997). However there are examples where an even shorter period [e.g. shift