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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.