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recently , increasing attention has been paid on the understanding of linguistic systems by computational and analytical methods . physicists , mathematicians , computer scientists and biologists apply their tools on the investigation of language capability , language change and language competition@xcite . especially the similarity to biological systems opens the field of linguistics to methods of the far better understood evolutionary systems . for an overview see refs.@xcite . it is believed that a concept of `` universal grammar '' is fixed by some way in our genetic code enabling humans to learn languages fast during our childhood@xcite . much attention has been paid on the evolution of this trait and the competition between different grammars@xcite . in these models the different grammars compete by giving higher fitness to better competitors . because of that it is possible to apply directly models already successfully used in evolutionary biology . the competition between languages has been investigated in the last years analyzing the stability of an system of two or more languages@xcite as well as their size distribution@xcite . in our approach we use a computational model _ without _ giving different fitness to agents with different language traits and without neglecting bilinguals . models of that kind can be found in refs.@xcite . this makes our model different from biological systems . the knowledge of a different language does not imply a higher death probability ( we are aware that , unfortunately , there are quite many exceptions of that rule ) . in order to present an age  structured model we build our one on the penna model@xcite . by the treatment of the genes like bits the model yields a rather realistic age structure of a sexually reproducing population , and is additionally suitable for large population sizes due to its low computational cost . the usage of an age  structured model enables us to merge an age dependent learning procedure with a model of language competition in order get more insight into their relation . we will focus on the question whether a phase transition exists , as found in the models of refs.@xcite , between the state of dominance of one language and an uniform distribution ( also called fragmentation ) . the paper is organized as follows : the next section explains the model , the following section presents the results of the model with respect to its parameter space . the last section discusses the results and proposes further extensions of the model . the model in this paper combines population genetics with age  dependent language learning . the presentation of the genetical part , with the purpose to yield a stable population having its characteristic age  structure , will be provided only shortly to the reader . for a more detailed description of the penna model and its applications we refer to refs.@xcite . [ [ the - sexual - penna - model ] ] the sexual penna model + + + + + + + + + + + + + + + + + + + + + +    is a individual  based model where every agent is represented by its age and two strings ( diploid ) of 32 bits . the bit  strings are read in parallel . every iteration after birth a new position on each of the bit strings becomes visible . a bit set to one corresponds to an harmful allele . at the beginning of a simulation five positions are randomly chosen to be dominant ones . if the position is dominant one bit of the two bit  strings suffices to make the position represent a deleterious mutation . on the other positions both bits need to be set to one for deleterious effects . at the age at which an agent has its third deleterious mutation it dies . in order to avoid population explosion an additional factor limits its growth : the verhulst factor @xmath0 gives the probability of an agent to survive an iteration . here @xmath1 is the current population size and @xmath2 the so  called carrying capacity defining the maximum size of a population . after reaching the reproduction age @xmath3 a female agent every iteration chooses randomly a male with age equal or older than @xmath4 in order to produce one offspring . for each parent its two bit  strings are randomly crossed and recombined . both parents contribute one string to the offspring . the offspring suffers one deleterious mutation at a randomly chosen position on each of its both new strings . this high mutation rate lets the population reach a stationary distribution fast . the two bit  strings are initialized randomly . generate a new language spoken by the child on a randomly chosen position.,scaledwidth=40.0% ]    [ [ the - language - trait ] ] the language trait + + + + + + + + + + + + + + + + + +    of an agent , that means its current ability to speak certain languages , is modelled in a similar way as the genetic strings of biological ageing . the whole structure , a language should consist of , is neglected , and language is treated as an unit . each agent contributes one additional bit  string of @xmath5 bits representing its capability to speak a maximum of @xmath5 languages . so an agent who has for instance its third bit set , speaks currently language @xmath6 ( eventually among others ) . an agent can learn or forget a language only in youth during the first @xmath7 iterations after birth . we call @xmath7 the _ maximum learning age_. older agents are not able to change their knowledge on languages . the interaction between a young agent and a randomly chosen teaching agent works as follows : at first a random position on the language bit  string is chosen . the agent learns the language of that position if it is spoken by the teacher . if the agent already speaks more than one language it forgets the language of the chosen position if the teacher does not know it . in the other cases no process of learning / forgetting is carried out . let us illustrate the outcome of the interaction in computational terms : the language trait of an agent speaking zero or one language has an or operation with the teacher s one at the chosen position , otherwise an and operation . the young agents s language traits are actualized in this way by @xmath8 different teachers at different positions per iteration . @xmath9 means that the language traits are actualized once per iteration with a probability of @xmath10 . at birth an offspring obtains its language trait as illustrated in figure  [ fig : illustrate ] . if both parents speak the same language the offspring will speak it as well . in the case that only one of the parents speaks a language the child learns it with a probability of @xmath10 . additionally , the child can know a new language at birth with probability @xmath11 ( i.e. a randomly chosen bit position is set to one ) . we call this probability @xmath11 of erroneous learning the _ mutation rate _ in the following because of its analogy to biological systems . the bit  string is initialized by chance at the beginning of a simulation . this section will concentrate at first on the results of simulations with a trait of two languages ( @xmath12 ) . the simulations are carried out for at least @xmath13 iterations or more for large population sizes . all results present the final stationary state of a simulation . a phase transition can be observed for the ratio between the number of speakers of different languages . the ratio is averaged over the last @xmath14 time steps . figure  [ fig : phasetr1 ] shows the ratio versus the mutation rate , determining the control parameter , for different carrying capacities @xmath2 . the simulations are carried out with a maximum learning age of @xmath15 and @xmath9 . each point represents the outcome of a separate simulation . the language ratio decreases fast at the critical point which is situated between @xmath16 and @xmath17 . the transition separates the phase / state of about equal sizes for the two languages for small values of @xmath11 from the phase / state where one language dominates clearly the other one . we can observe in figure  [ fig : phasetr1 ] as well that the population size does not alter the shape of the transition . thus this transition can be found for arbitrary population numbers and therefore corresponds to a real physical phase transition . the language ratio increases strongly for low mutation rates . a power law with an exponent of minus one is exhibited in figure  [ fig : powerlaw ] . this exponent is the same for all simulations presented here . the parameters in the simulations are @xmath15 , @xmath9 and @xmath18 , i.e. the population consists of about @xmath19 agents .     as well as with increasing amount @xmath8 of interaction per iteration.,scaledwidth=80.0% ]    in figure  [ fig : phase_interact ] the shape of the phase transition is compared for different values of @xmath7 and @xmath8 , parameters defining the amount of interaction between the agents . the carrying capacity @xmath20 fixes the population number at around @xmath21 . the shape of the transition remains the same but is shifted to higher values of the mutation rate for increasing @xmath7 and @xmath8 . the shift is a linear one with respect to each of both parameters . the shift decreases for values of @xmath22 as expected due to the smaller number of individuals surviving up to ages much older than the reproduction age @xmath3 ( not shown ) . the phase transition disappears and the only final state of the model is the dominant one for very large values of @xmath8 .      in the following the results of simulations with a language trait consisting of more than two languages are compared to the previous ones . the language ratio @xmath4 , the measure for a possible domination of one over the others , is now defined by @xmath23 where @xmath24 is the most spoken language , @xmath5 the number of languages , and @xmath25 the number of agents speaking language @xmath26 . the simulations are carried out with @xmath27 and @xmath20 . we used @xmath28 for @xmath29 , respectively in order to keep the interaction per language constant . figure  [ fig : numwords]a compares the shape of the transition for different numbers of languages . the position of the critical point is shifted to smaller values of @xmath11 for more languages . this means that the more languages compete against each other , the lower is the possibility to have a scenario where one language dominates the other ones . figure  [ fig : numwords]b compares the same simulations carried out with the initial state where all agents speak language @xmath30 . the critical point is shifted to larger values of the mutation rate . this dependence of the final state on the initialization , leading to a hysteresis , shows that the phase transition is of first order . here the jump from one phase into the other is located at the intersection of the transition curve and the curve for the ratio of two languages .     languages . the critical point moves to smaller values of the mutation rate for increasing @xmath5 . the phase transition is more abrupt . * a * : for numbers of languages larger than two the curves collapse into one for values of @xmath11 in the left part where one language dominates the other ones . * b * : the initialization with one language leads to a different result . the transition shows a hysteresis for the simulations with more than two languages increasing with @xmath5.,title="fig:",scaledwidth=70.0% ]   languages . the critical point moves to smaller values of the mutation rate for increasing @xmath5 . the phase transition is more abrupt . * a * : for numbers of languages larger than two the curves collapse into one for values of @xmath11 in the left part where one language dominates the other ones . * b * : the initialization with one language leads to a different result . the transition shows a hysteresis for the simulations with more than two languages increasing with @xmath5.,title="fig:",scaledwidth=70.0% ]    finally , we present a histogram of the hamming distances@xcite between all agents for simulations with 16 languages for different values of the mutation rate ( figure  [ fig : hamming ] ) . the parameters are the same as before . the hamming distance is defined by the number of bits by which the two bit  strings differ from each other . or , in other words , the number of bits which need to be changed to turn the language trait of an agent into that of the other one . figure  [ fig : hamming ] shows that the peak of the histogram of hamming distances turns from zero ( dominance ) to one ( no dominance ) at the critical point ( which here is located between @xmath31 and @xmath32 ) . in our model of language competition a phase transition similar to the one for the competition of different grammars in ref.@xcite and the one for the competition of languages of ref.@xcite is observed . nevertheless , working without fitness improvement for agents better adapted by their languages trait but with a sexual population with random mating makes the simulations useful to characterize current language competition . the phase of an uniform distribution of the languages ( fragmentation ) suggests a scenario where these languages compete equally against each other maintaining a stable state of coexistence . in the other state one language dominates clearly the other ones . the dependence of the phase transition point on the amount of language interaction as well as on the number of competing languages lets us make the following conclusions : there is no optimal value of maximum learning age for a language . its increase shifts the critical point to larger values of the mutation rate . thus a larger maximum learning age prevents the evolution of many languages where contact among speakers of different languages is strong . the model shows also that a larger number of different languages decreases the possibility of dominance of one of them and makes the transition jump higher . hence , the extinction of a language in a multilingual system can lead to a transition to the state where one language will begin to dominate all the other ones . the phase transition analyzed in the presented model is of first order for simulations with many languages as found in the model of ref.@xcite . the differences are that in our case , the transition is completely independent of the population size . we do not know if the simplification of a language to a single bit leads to this discrepancy or if it is the usage of a model with sexual reproduction . further analysis has to be done . the model brought insight into the mechanisms of language competition but still much has to be done . the results are similar to the ones of ref.@xcite . nevertheless , our model presents another approach to understand language competition . the design of an analytical model , for instance by diffusion equations similar to the ones used in ref.@xcite , without fitness would reveal if it is possible to obtain similar results with a mean field theory . the model presented in this paper can be extended by putting it on a lattice in order to investigate how the transition behaves with respect to a geographic distribution of languages . another extension would be the substitution of random mating by assortative mating@xcite , a concept used frequently in the theory of biological speciation , in order to give different priorities to monolingual parents and bilingual ones . in order to understand language invasion , the model can be extended to include social structures . i am funded by the daad ( deutscher akademischer austauschdienst ) and thank d. stauffer and s. moss de oliveira for comments on my manuscript .
the understanding of language competition helps us to predict extinction and survival of languages spoken by minorities . 
 a simple agent  based model of a sexual population , based on the penna model , is built in order to find out under which circumstances one language dominates other ones . 
 this model considers that only young people learn foreign languages . 
 the simulations show a first order phase transition where the ratio between the number of speakers of different languages is the order parameter and the mutation rate is the control one .