Page 126 - Tallinn University

that births, deaths and migration cancel each-other out, thus the total population size does not
= !
!! !!! +1+ (8) (!!!) ! +1+ ! ! !
=
! ! ! 1− − (!!!)
1− − !!! !!!! !1− − !!! !!!!
change over time. The society is then made up of agents (speakers) of linguistic groups X, Y
Tallinna Ülikooli üliõpilaste 2015/2016. õppeaasta PARIMAD TEADUSTÖÖD / Artiklite kogumik HARIDUSTEADUSED and Z, who change their respective group based on a transition probability function P. The
latter describes the attractiveness of a language to the population, determining the general
outcome of the competition. The three-language model is written:
e three-language model is written:
= , + , − , − , ,
= , + , − , − , , (1) !" ! !" ! !" ! !" !
= , + , − , − , , !" ! !" ! !" ! !" !
where x, y and z are the fraction of the total population belonging to linguistic group X, Y and Z, where x, y and z are the fraction of the total population belonging to linguistic group X, Y and
respectively. is can also be viewed as a probability that a random agent encountered in this system
Z, respectively. This can also be viewed as a probability that a random agent encountered in
will either speak the language X, Y or Z.
this system will either speak the language X, Y or Z.
Z, respectively. This can also be viewed as a probability that a random agent encountered in Z, respectively. This can also be viewed as a probability that a random agent encountered in
P is the transition probability function, where the direction of change is denoted with a subscript, read this system will either speak the language X, Y or Z.
P is the transition probability function, where the direction of change is denoted with a
this system will either speak the language X, Y or Z.
frosmublsec ritpotr, irgehatd. Sfirnocme tlreafnt stiotioringhptr.oSbianbcielittyradnespiteinodnspornolbyaobniltithye dpeapraemndesteornsloyf othnetrheecipaieranmt leintegrusistic P is the transition probability function, where the direction of change is denoted with a
group Y or Z, when swapping to X, the probabilities and are equal (this relation holds
P is the transition probability function, where the direction of change is denoted with a of the recipient linguistic group, then regardless of whether an agent previously belonged to
group, then regardless of whether an agent previously belonged to group Y or Z, when swapping to X,
subscript, read from left to right. Since transition probability depends only on the parameters subscript, read from left to right. Since transition probability depends only on the parameters
true for any transition, respectively):
group Y or Z, when swapping to X, the probabilities and are equal (this relation holds
!" !"
group Y or Z, when swapping to X, the probabilities and are equal (this relation holds
theopf rtohbearbeiclitpieiesnat nlidngauriesteiqcugarlo(uthpi,sthrelnatrioegnahrdoledsstroufewfohretahneyr tarnanasgiteionnt ,presvpioecutsilvyelbye):longed to of the recipient linguistic group, then regardless of whether an agent previously belonged to
true for any transition, respectively): = = .
Volatility separates the system into two distinct states. When a ˂ 1, the system is considered values between 0 and infinity.
high volatility, characterised by a frequent change of linguistic group by agents. If volatility a > 1, then to be of high volatility, characterised by a frequent change of linguistic group by agents. If Volatility separates the system into two distinct states. When a ˂ 1, the system is considered
Volatility separates the system into two distinct states. When a ˂ 1, the system is considered thevosylastielmityisac˃on1s,idthereendtthoebseyostfelmowisvocloantislitdye,rcehdartoacbteriosfedlobwy mvolraetiilniteyr,t cahgaernatcst,esrwisaepdpibnyg mlaonrgeuage
to be of high volatility, characterised by a frequent change of linguistic group by agents. If to be of high volatility, characterised by a frequent change of linguistic group by agents. If
inert agents, swapping language less frequently. lesvsofrlaeqtiulietyntaly.˃1,thenthesystemisconsideredtobeoflowvolatility,characterisedbymore
volatility a ˃ 1, then the system is considered to be of low volatility, characterised by more inert agents, swapping language less frequently.
Prestige represent the relative status of a given linguistic group, representing the part of the inert agents, swapping language less frequently.
Prestige represent the relative status of a given linguistic group, representing the part of the society
society holding a favourable view of sa id+gr ou+p, th=us be. ing fractional and summing up to 1:
society holding a favourable view of said group, thus being fractional and summing up to 1:
Prestige represent the relative status of a given linguistic group, representing the part of the Prestige represent the relative status of a given linguistic group, representing the part of the
holding a favourable view of said group, thus being fractional and summing up to 1:

+ + = . (3) 1.1 Fixed points of the model
(3) society holding a favourable view of said group, thus being fractional and summing up to 1:
!" !"
!" !"

= = . (2)
(2) true for any transition, respectively): = = . (2)

In Eq. (2), a is the volatility parameter of the system and is the prestige (in the example case, indicated In Eq. (2), a is the volatility parameter of the system and is the prestige (in the example
In Eq. (2), a is the volatility parameter of the system and is the prestige (in the example !
In Eq. (2), a is the volatility parameter of the system and is the prestige (in the example deoscfrinbitnergestht eforrateheatmwohdieclh, ldaensgcuriabgiensgatrheeswraateppaetdwahnidchhloawng!iunaegrtesthaeresoscwieatpypiesdtoanladnhguowageinsehrit . It
by subscript, of linguistic group X). Volatility is one of the main parameters of interest for the model,
case, indicated by subscript, of linguistic group X). Volatility is one of the main parameters
case, indicated by subscript, of linguistic group X). Volatility is one of the main parameters
case, indicated by subscript, of linguistic group X). Volatility is one of the main parameters the society is to language shift. It defines the shape of the probability function and holds
de nes the shape of the probability function and holds values between 0 and in nity.
of interest for the model, describing the rate at which languages are swapped and how inert of interest for the model, describing the rate at which languages are swapped and how inert
values between 0 and infinity.
the society is to language shift. It defines the shape of the probability function and holds the society is to language shift. It defines the shape of the probability function and holds
Volatility separates the system into two distinct states. When a < 1, the system is considered to be of values between 0 and infinity.
!
+ + = . (3)
1.1 Fixed points of the model
The system has five fixed points, of which three are trivial and given by the following sets of
1.1 Fixed points of the model = = 1; = 0; = 0 , (4) 1.1 fixed pOintS Of the mOdel
!!!!
coordinates:
The system has five fixed points, of which three are trivial and given by the following sets of The system has five fixed points, of which three are trivial and given by the following sets of
coordinates: coordinates:
= =1; =0; =0 , (4) !!!!!!!!
= =1; =0; =0 , (4) != !=0; !=1; !=0 , (5)
e system has ve xed points, of which three are trivial and given by the following sets of coordinates:
= =0; =1; =0 , (5) != !=0; !=0; !=1 , (6)
= =0; =1; =0 , (5) !!!!!!!!
= =0; =0; =1 , (6)
!!!!
= =0; =0; =1 , (6)
!!!!
These correspond to any one language being the sole survivor. The fourth and fifth points,
aensedc or,recsoprorensdptoondantyooanesiltaunagtiuoangewbheirnegmthoeresotlheasnurovniveolra.n gueafgoeurhtahvaenadct i veh sppoeinaktse,rsa.nTdh,ecor-
!
These correspond to any one language being the sole survivor. The fourth and fifth points,
!
These correspond to any one language being the sole survivor. The fourth and fifth points,
! resapnodnd to,caosrirteusaptiondwtohearesmituoarteiotnhawnhoenrelmanogrueatgheahnaovneeacltainvgeusapgeeakhearvs.e aceticvoeosrdpeinaakterss.foTrhtheese
! and , correspond to a situation where more than one language have active speakers. The
!!
coordinates for these points are as follows:
coordinates for these points are as follows:
points are as follows:
coordinates for these points are as follows:
! !
= = !!!! !!!!!!!!!!
; = !!!! ; =0 , (7) !!! !!!
!!! !!!
!+ !+
= = ! !!!! ; = ! !!!! ; =0 , (7) !!!!!!!!
= = ! !!! ! ; = ! !!! ! ; =0 , (7) ! ! ! + ! ! ! + ! !
!!! !!! 1 !!! !!! !! !!
!!! !!! !!! !!! + +
! =!!!;
1 − − ! ! !1 ! ! ! !!!! !
= 1+1+! ;
= ; !!!!!!
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