The impo ance o in e linguis ic simila i y and
s able bilingualism when wo languages compe e
To ci e his a icle: J Mi a e al 2011 New J. Phys. 13 033007
View he a icle online o upda es and enhancemen s.
Rela ed con en
Agen based models o language
compe i ion: mac oscopic desc ip ions and
o de –diso de ansi ions
F Vazquez, X Cas elló and M San Miguel
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Coupled dynamics o node and link s a es
in complex ne wo ks: a model o language
compe i ion
Ad ián Ca o, Raúl To al and Maxi San
Miguel
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In e linguis ic simila i y and language
dea h dynamics
J. Mi a and Á. Pa edes
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Recen ci a ions
Using complex ne wo ks o quan i y
consis ency in he use o wo ds
D R Amancio e al
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The open–access jou nal o physics
New Jou nal o Physics
The impo ance o in e linguis ic simila i y and
s able bilingualism when wo languages compe e
J Mi a1,3, L F Seoane1and J J Nie o2
1Depa amen o de Física Aplicada, Uni e sidade de San iago de Compos ela,
15782 San iago de Compos ela, Spain
2Depa amen o de Análise Ma emá ica and Ins i u o de Ma emá icas,
Uni e sidade de San iago de Compos ela, 15782 San iago de Compos ela,
Spain
E-mail: jo [email p o ec ed]
New Jou nal o Physics 13 (2011) 033007 (9pp)
Recei ed 6 Oc obe 2010
Published 3 Ma ch 2011
Online a h p://www.njp.o g/
doi:10.1088/1367-2630/13/3/033007
Abs ac . One app oach o analyzing he dynamics o wo languages in
compe i ion is o i his o ical da a o he numbe o speake s o each wi h a
ma hema ical model in which he pa ame e s a e in e p e ed as he simila i y
be ween hose languages and hei ela i e s a us. Wi hin his app oach, on he
basis o a de ailed analysis and ex ensi e calcula ions, we show he ou comes ha
can eme ge o gi en alues o hese pa ame e s. In con as o p e ious esul s,
i is possible ha in he long e m bo h languages may coexis and su i e. This
happens only whe e he e is a s able bilingual g oup, and his is possible only i
he compe ing languages a e su icien ly simila , in which case i s occu ence is
a ou ed by bo h simila i y and s a us symme y.
Con en s
1. In oduc ion 2
2. Me hods and esul s 3
3. Final ema ks and u u e wo k 7
Acknowledgmen 9
Re e ences 9
3Au ho o whom any co espondence should be add essed.
New Jou nal o Physics 13 (2011) 033007
1367-2630/11/033007+09$33.00 © IOP Publishing L d and Deu sche Physikalische Gesellscha
2
1. In oduc ion
An aspec o globaliza ion ha ala ms many is he eplacemen o local ongues by mo e
hegemonic languages [1], a end ha has been in es iga ed om mul iple poin s o iew,
including ha o physics [2]–[7]. Whe e wo languages compe e, i is easonable o assume
ha he ou come is in luenced by hei ela i e s a us [6] (i.e. he speake s’ pe cep ions o
he social and/o economic ad an ages each language o e s) and hei simila i y [7] (o he
complemen a y concep , in e linguis ic dis ance [8,9]), bu conside a ion o hese ac o s has
been hinde ed by di icul ies in hei ope a ionaliza ion.
One o he ea lies and simples ma hema ical models o language shi was ha o Ab ams
and S oga z [6], [10]–[12], who conside ed a s able popula ion in which wo languages wi h
di e en s a uses compe ed o speake s. This model, which in ol es analysis o he e olu ion
o he numbe o speake s o e ime, p edic ed ha one o he languages would ine i ably die
ou , and was success ully i ed o his o ical da a on he compe i ion be ween Sco ish Gaelic
and English, Welsh and English, and Quechua and Spanish, among o he language pai ings [6].
Howe e , i did no ake in o accoun he possibili y o bilingual indi iduals, a possibili y ha
is o cou se ealized in nume ous mul ilingual socie ies. In Spain, o example, whe e Cas ilian
Spanish is he o icial language h oughou he s a e, bu in ce ain egions is co-o icial wi h
ano he language (mainly Galician, Basque, Ca alan o Valencian), indi idual bilingualism is
common in communi ies wi h mo e han one o icial language.
We ecen ly showed ha he his o ical e olu ion o he use o Galician and Cas ilian in
Galicia (NW Spain) can be explained by a modi ied Ab ams–S oga z model ha allows o
bilingual as well as monolingual speake s o he compe ing languages, and ha includes a
pa ame e ha ep esen s he ease o bilingualism [7] ( igu e 1). We conside ed a popula ion
in which monolingual speake s o he language X make up a ac ion xo he popula ion,
monolingual speake s o he language Y accoun o a ac ion yand he bilingual (B) o
a ac ion b(wi h x+y+b=1). Th oughou his pape , capi al le e s X and Y deno e he
wo languages spoken in he popula ion; he uppe case le e B deno es he g oup o bilingual
speake s; and he lowe case le e s x,yand b e e o he ac ion o speake s o each o he
languages in he popula ion and he ac ion o bilingual speake s, espec i ely.
The dynamics o language change a e acco dingly desc ibed by he sys em
dx
d
=y PYX +bPBX −x(PXY +PXB), (1a)
dy
d
=x PXY +bPBY −y(PYX +PYB), (1b)
db
d
=x PXB +y PYB −b(PBY +PBX), (1c)
whe e PXY deno es he p obabili y o a monolingual speake o language X being eplaced in
he popula ion by a monolingual speake o language Y, wi h analogous no a ion o he o he
possible eplacemen s. The p obabili y o a monolingual pe son being eplaced by a mono- o
bilingual speake o he o he language is assumed o be p opo ional bo h o he s a us o he
second language, i.e. he social and/o economic ad an ages i o e s, and o a powe o he
p opo ion o he popula ion ha speaks i . Thus, deno ing by s he ela i e s a us o language X
New Jou nal o Physics 13 (2011) 033007 (h p://www.njp.o g/)
3
P(YX)∝1-k
P(XY)∝1-k
X-Speake s Y-Speake s
Bilinguals
P(YB)∝k
P(XB)∝k
Figu e 1. Flux among X-monolingual, Y-monolingual and bilingual g oups.
Acco ding o he model se o h in equa ions (1a)–(1c) and (2a)–(2d), lux is
go e ned bo h by he ela i e s a uses o he compe ing languages ( he ela i e
social and/o economic ad an ages hey o e ) and by in e linguis ic simila i y k,
he p obabili y ha he disappea ance o a monolingual speake o one language
will be compensa ed o by he appea ance o a bilingual a he han by a
monolingual speake o he o he language.
and by 1 −s ha o language Y,
PX B =c·k(1−s)(1−x)a,(2a)
PY B =c·ks(1−y)a,(2b)
PB X =PY X =c·(1−k)s(1−y)a,(2c)
PBY =PXY =c·(1−k)(1−s)(1−x)a,(2d)
whe e cis a no maliza ion ac o ela ed o he ime scale, ais he powe pa ame e and kis he
p obabili y ha he disappea ance o a monolingual speake o language X ( espec i ely Y) will
be compensa ed o by he appea ance o a bilingual a he han by a monolingual speake o
language Y ( espec i ely X). We iden i y in e linguis ic simila i y wi h his pa ame e k.
No e ha since people do no ac ually o ge hei na i e ongue(s), he abo e model is a
model o popula ion enewal, a he han o indi iduals swi ching om one linguis ic p ac ice
o ano he , excep ha i also allows hose who a e cu en ly monolingual o become bilingual.
When k=0, i educes o he Ab ams–S oga z model [6] o decays owa ds his model i bis
ini ially non-ze o. No e also ha popula ion g ow h does no in alida e he model, so long as
he a ious linguis ic g oups a e all a ec ed in p opo ion o hei size.
Al hough he model ske ched abo e adequa ely accoun s o he Galician da a up o he
p esen [7], he ques ion a ises whe he his si ua ion is s able. Mo e gene ally, wha a e he
possible long- e m ou comes o compe i ion be ween wo languages, and unde wha condi ions
do hey come abou ? Namely, migh wo languages coexis s ably?
2. Me hods and esul s
To in es iga e hese issues we ha e ca ied ou ex ensi e calcula ions, sys ema ically a ying s
and k, o de e mine popula ion s a es (x,y) ha ac as poin a ac o s o he coupled sys em
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Figu e 2. E olu ion o language dominance when languages X and Y compe e.
(a) The p opo ion o ini ial speake s o he languages X and Y is ep esen ed
by a poin in he g aph, each poin in his panel being a possible ini ial
dis ibu ion gene a ed a andom. (b–d) Each ini ial dis ibu ion o speake s
e ol es acco ding o he sys em o equa ions (1a)–(1c). The e olu ion o
he p opo ion o speake s o a gi en ini ial si ua ion is acked down by
he ajec o y o he poin s h ough he phase space. He e we p esen he
e olu ions a e 100, 300 and 1500 s eps o he compu a ion, espec i ely. In
he example s=0.75 and k=0.3. Wi h hese pa ame e s, he inal dis ibu ion
o speake s will be ha wi h all he popula ion being monolingual wi h
language X.
de ined by equa ions (1a)–(1c) and (2a)–(2d) wi h x+y+b=1; he p oblem is well posed
because, as is easily shown, he eloci y ield on he bounda y o he se o possible s a es
(de ined by x⩾0, y⩾0, x+y⩽1) ne e akes he sys em ou side his se , wha e e he alues
o sand k.
Ini ially, ou calcula ions we e pe o med as ollows. Fo all (s,k)o he o m
(0.05ns,0.05nk) (0 ⩽ns,nk⩽0), 10 000 s a ing s a es (x0,y0)we e andomly selec ed in he
se o possible s a es, and he disc e ized o m o he abo e sys em (1a)–(1c) was sol ed
nume ically o each, wi h a=1.31, using a ime s ep 1 =0.01 ( he alue a=1.31 was
chosen because Ab ams and S oga z [6] ound ha his pa ame e was su p isingly cons an ,
1.31 ±0.25, when hei model was i ed o 42 da a se s conce ning dissimila language pai s).
New Jou nal o Physics 13 (2011) 033007 (h p://www.njp.o g/)
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Figu e 3. A case wi h wo di e en a ac o s. Again, andom p opo ions o
speake s o each language we e p o ided as ini ial condi ions o equa ions
(1a)–(1c), his ime wi h pa ame e s s=0.67 and k=0.77. The sampled poin s
e ol e acco ding o hese equa ions and snapsho s o he phase space a e aken
a e (a) 200, (b) 700, (c) 6000 and (d) 10 000 compu a ion s eps. In his
case, all he ini ial dis ibu ions o speake s e ol ed owa ds a small egion,
which is zoomed-in in panels (c) and (d). Fo hese pa ame e s, and dep-
ending on he ini ial p opo ions o he X-monolingual, he Y-monolingual and
he bilingual, he language Y ei he dies ou comple ely o su i es, in he la e
case mos ly among bilinguals. No e he opening o a gap in he poin s o panel
(d), because he s a es e ol e owa ds di e en a ac o s (x=1,y=0)and
(x=0.6,y=0.04).
The 10 000 calcula ions p oceeded concu en ly and we e hal ed as soon as all he s a e poin s
had con e ged o wi hin a ci cle o adius 10−5. Figu es 2(a)–(d) show selec ed s ages o his
p ocess o he case (s=0.75, k=0.3). Howe e , i was soon ound ha he e we e alues o
(s,k) o which he s a e poin s did no con e ge, and examina ion o hei dis ibu ion indica ed
ha his was due o he exis ence o mo e han one poin a ac o , he a ac o o which any
pa icula s a e poin ended, depending on i s s a ing s a e. In hese cases, he calcula ions
we e allowed o p oceed o imes se e al o de s o magni ude longe han he a e age single-
a ac o con e gence ime, un il all he s a e poin s we e wi hin wo o h ee ci cles o adius
10−5( igu e 3shows he esul o he case s=0.67, k=0.77).
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Figu e 4. The i e possible s able si ua ions de e mined by sand k. In panels
(a–e), he poin s ep esen ing he ini ial andom condi ions o i e simula ions a e
classi ied acco ding o he a ac o owa ds which hey con e ge. The di e en
a ac o s a e ep esen ed by a huge black spo . All he ini ial dis ibu ions o
speake s ha collapse o he same s able si ua ion sha e he same colou . The
pa ame e s kand sde e mine he numbe and posi ion o he a ac o s and allow
a so ing o he o e all sys em in o i e opologically dissimila ypes (see he
ex o hei desc ip ion). In panel (a), an example o ype I is shown (s=0.80,
k=0.65); (b) ype II (s=0.40, k=0.20); (c) ype III (s=0.50, k=0.65);
(d) ype (IV) (s=0.35, k=0.75); and (e) ype (V). In ( ) his so ing is made
explici . All pai s o pa ame e s leading o he same opological dis ibu ion o
a ac o s a e colou ed oge he . A whi e spo localizes pa ame e s kand so he
sys em Galician–Cas ilian.
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Depending on sand k, he ollowing i e si ua ions we e obse ed o eme ge.
(I) The e is jus a single s able s a e a x=1 o y=1, i.e. one o he languages becomes
ex inc e en hough i may ini ially ha e been dominan . This is he beha iou illus a ed
in igu es 2(a)–(d) and in igu e 4(a).
(II) The e a e s able s a es a bo h x=1 and y=1; one o he languages dies ou , bu which
one depends on he ini ial dis ibu ion (x0,y0), as illus a ed in igu e 4(b).
(III) The e a e s able s a es a x=1 and y=1, oge he wi h a hi d one ha lies below he line
x+y=1 and hus co esponds o he p esence o a s able bilingual g oup ( igu e 4(c)).
(IV) The e is a s able s a e lying below he line x+y=1 (i.e. wi h a non-emp y bilingual
g oup), oge he wi h jus one s able monolingual s a e a x=1 ( igu es 3(a)–(d)) o y=1
( igu e 4(d)).
(V) The e is jus one s able s a e, and i includes a s able bilingual g oup ( igu e 4(e)).
No e ha in cases (II), (III) and (IV) he e a e sha p bounda ies be ween he ini ial s a e zones
leading o di e en ou comes; a hese bounda ies, an exogenous injec ion o jus a ew speake s
in o one g oup o ano he can de e mine whe he a language li es o dies, as is illus a ed in
igu es 5(a) and (b).
Figu e 4( ) shows which o he abo e i e si ua ions each (s,k) alue led o. The salien
aspec s o his map a e ha he s able exis ence o a bilingual g oup equi es ha kexceed a
minimum alue o abou 0.35; ha i k⩾0.6, hen he less symme ic languages X and Y a e
s a us-wise, he la ge kmus be o s able bilingualism; and ha when he wo languages a e
mode a ely symme ic s a us-wise, which o hem disappea s depends on he ini ial sizes o he
linguis ic g oups i hey a e essen ially dissimila (k⩽0.4), bu no i hey a e mo e simila
(0.4<k<0.6).
Re u ning o he case o Galician and Cas ilian in Galicia, he inclusion o ecen ly
published da a [13] in he analysis ( igu e 5(c)) co obo a es ou p e ious es ima es [7] o
sGalician (0.26) and k(0.80), in ai ly good ag eemen wi h lexicos a is ics-based es ima es o
simila i y among closely ela ed Romance languages [14]. These alues place Galicia in zone I
o igu e 4( ), and acco dingly p edic he e en ual ex inc ion o Galician. Howe e , he close
p oximi y o zone IV sugges s, and igu e 5(c) shows, ha ex inc ion is no imminen : in ac ,
i is p edic ed ha by he end o he cen u y he popula ion will be oughly equally di ided
be ween monolingual Cas ilian speake s and he bilingual.
3. Final ema ks and u u e wo k
A e exhaus i e esea ch in espec o he p oposed model, we can conclude ha ma hema ical
solu ions showing he su i al o bilingual speake s a e possible, and ha hese solu ions a e
also linked o he su i al o g oups o monolingual speake s o bo h compe ing languages. This
s eady coexis ence o all he linguis ic g oups can depend on he global pa ame e s o he sys em
(bo h languages being mo e likely o su i e whe e he e is la ge in e linguis ic simili ude and
a s a us close o 0.5), bu also on he ini ial popula ion suppo ing each ongue when he sys em
is go e ned by ce ain se s o pa ame e s (k,spai s in zones III and IV in igu e 4).
The model used in his pape has wo e iden limi a ions. Fi s ly, he model does
no conside possible al e a ions o he ela i e p opo ions o he linguis ic g oups due o
immig a ion, emig a ion o di e en ial bi h and/o dea h a es. Secondly, pa ially ela ed o
New Jou nal o Physics 13 (2011) 033007 (h p://www.njp.o g/)
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Figu e 5. Time e olu ion o single cases. Time dependence o he p opo ions
o he X-monolingual ( ed), he Y-monolingual (g een) and he bilingual (blue).
Panels (a) and (b) illus a e o a ype II sys em (s=0.53, k=0.55) how he a e
o he sys em can depend c i ically on ini ial condi ions, language Y becoming
ex inc i ini ially x=0.84 and y=0.15 (a), bu no i x=0.83 and y=0.15 (b).
(c) Resul s o i ing he model o his o ical da a o Galician (X, ed), Cas ilian
(Y, g een) and he bilingual (B, blue) in Galicia (NW Spain). Al hough he i ed
alues o sand k, 0.26 and 0.80, place Galicia in zone I o igu e 4( ), hus
p edic ing he e en ual ex inc ion o Galician, hose bilingual in Galician and
Cas ilian a e no expec ed o disappea wi hin his cen u y o he nex .
his is he conside a ion ha he ela i e s a us o he wo languages may well a y in ime.
Also, o cou se, i emains o be seen whe he he de ini ion o linguis ic simila i y in e ms o
he dynamics o he popula ion enewal p ocess co esponds o any pu ely linguis ic concep ;
he assump ion ha simila i y is symme ic, i.e. ha PXB/[(1−s)(1−x)a]=PYB/[s(1−y)a],
is clea ly an idealiza ion; and he e is he issue o whe he wo e y simila languages eally
ha e sepa a e iden i ies, especially when hey ha e a simila s a us and one o bo h only su i e
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