Modeling Language CompetitionModeling Language Competition

Midterm Status Report

April 2010

University of Arizona

Erin Hittle, Katie Goeldner, Brian Bilquist, and Samantha Vandermey

Mentored by: Rebecca Stockbridge

IntroductionIntroduction

� By looking at proportions and populations of two geographically equivalent languages we are attempting to determine the dynamics of language to determine the dynamics of language interaction.

� Interested in determining whether or not coexistence is possible

ExamplesExamples

� French vs. English in Canada

� Scottish and Irish Gaelic vs. English

� Welsh vs. English in Wales� Welsh vs. English in Wales

� Quechua vs. Spanish in Peru

Modeling the dynamics Modeling the dynamics of language deathof language deathBy Daniel M. Abrams and Steven H. Strogatz

� Languages are fixed and compete with each other for speakers

� A highly connected population with no � A highly connected population with no spatial or social structure in which all speakers are monolingual

� Attractiveness of language increases with number of speakers and perceived status

Modeling the dynamics of language deathModeling the dynamics of language death

Given competing languages X and Y,

where

� - proportion of Y speakers converted to X

),(),( sxxPsxyPdt

dxxyyx −=

),( sxyP� - proportion of Y speakers converted to X

� - proportion of X speakers converted to Y

and

scxsxP a

yx =),(

)1()1(),( sxcsxP a

xy −−=

),( sxyPyx),( sxxPxy

c – constant derived from data a – constant derived from datas – perceived status of language x

Modeling the dynamics of language deathModeling the dynamics of language death-- Conclusions and issuesConclusions and issues

� Three fixed points at x=0, x=1, and 0≤x≤1

◦ Only x=0 and x=1 are stable

◦ Indicates two languages cannot coexist in the long runlong run

� However, bilingual societies do exist –though they generally involve split populations without significant interaction.

Microscopic AbramsMicroscopic Abrams--StrogatzStrogatz model model of language competitionof language competitionBy Stauffer, By Stauffer, CastelloCastello, , EquiluzEquiluz, and Miguel, and Miguel

XYYX xPPxdt

dx−−= )1(

Where:asxP = Is the probability of switching from language

Where:a

YX sxP = Is the probability of switching from language Y to X

• a>1 same dynamics as original paper• a<1 stability flips• a=1 qualitative behavior similar

Microscopic AbramsMicroscopic Abrams--StrogatzStrogatz model model of language competition of language competition -- DynamicsDynamics

s = 0.8a = 1

Microscopic AbramsMicroscopic Abrams--StrogatzStrogatz model model of language competition of language competition -- DynamicsDynamics

s = 0.5

t

xs = 0.5a = 1

Microscopic AbramsMicroscopic Abrams--StrogatzStrogatz model model of language competition of language competition

-- Conclusions and issuesConclusions and issues

� Coexistence only possible if languages have equal prestige or if a<1

� No existence of bilingual population

� Social structure affecting prestige very simple

Coexistence of Languages is PossibleCoexistence of Languages is Possible

By J.P. Pinasco and L. Romanelli

� Languages x and y are spoken by population X(t) and Y(t)population X(t) and Y(t)

� x is the only attractive language

� c is the rate of conversion from y to x

� are parameters adjusting for natality and mortality rates

,xα yα

Coexistence of Languages is PossibleCoexistence of Languages is Possible

� Growth rates given by

� Then to model language competition, we

,1 and 1

Υ−Υ

Χ−Χ

y

y

x

x

SSαα

� Then to model language competition, we

get the following differential equations:

−+−=

−+=

y

y

x

x

S

YYcXY

dt

dY

S

XXcXY

dt

dX

1

and 1

α

α

Coexistence of Languages is PossibleCoexistence of Languages is Possible� Fixed points are found to be

� A fourth equilibrium point is found when the threshold condition is satisfied:

( ) ( ) ( )0, ,,0 ,0,0 xy SS

S < αy

◦ The carrying capacity Sx is small and it is easily reached.

◦ The rate of growth of population Y is high.

◦ There is a low rate of shift from the language y to x.

Sx < αy

c

Coexistence of Languages is Coexistence of Languages is Possible Possible -- DynamicsDynamics� With no conversion (c = 0),

10

10

=

=x

αα

600

500

10

=

=

=

y

x

y

S

S

α

Coexistence of Languages is Coexistence of Languages is Possible Possible -- DynamicsDynamics

•With Threshold Condition,

10

005.0

=

=c

α

600

500

10

10

=

=

=

=

y

x

y

x

S

S

αα

Coexistence of Languages is Coexistence of Languages is Possible Possible -- DynamicsDynamics� Without Threshold Condition

10

008.0

=

=

x

c

α

600

1500

10

10

=

=

=

=

y

x

y

x

S

S

αα

Compare and ContrastCompare and Contrast

� Abrams and Strogatz◦ Proportion◦ One dimensional◦ Motivation to switch to either language◦ Coexistence in very particular circumstances

Coexistence is PossibleCoexistence in very particular circumstances

� Coexistence is Possible◦ Population◦ Two dimensional◦ x is the only attractive language◦ Incorporates environment◦ Coexistence in a broader range of conditions

Further ImprovementsFurther Improvements

� Dialects

� Multiple language interactions

� Language Mutation

� Incorporation of bilingual populations� Incorporation of bilingual populations

◦ Introduce a third variable

◦ Continuous or discrete time

ReferencesReferences

� J. Magnet, Consitutional Law of Canada: Quebec’s Grievances and Proposals. (2002)

� D.M. Abrams, S.H. Strogatz, Nature 424 � D.M. Abrams, S.H. Strogatz, Nature 424 (2003) 900.

� D. Stauffer, X. Castello, V. Equiluz, M. San Miguel, Physica A 364 (2007) 835-842.

� J.P. Pinasco, L. Romanelli, Physica A 361 (2006) 355-360.

Questions?Questions?