We investigate the dynamics of two agent based models of
language competition. In the first model, each individual can be in
one of two possible states, either using language $X$ or language
$Y$, while the second model incorporates a third state $XY$,
representing individuals that use both languages (bilinguals).
We analyze the
models on complex networks and two-dimensional square lattices by
analytical and numerical methods, and show that they exhibit a
transition from one-language dominance to language coexistence. We
find that the coexistence of languages is more difficult to maintain
in the Bilinguals model, where the presence of bilinguals in use
facilitates the ultimate dominance of one of the two languages. A
stability analysis reveals that the coexistence is more unlikely to
happen in poorly-connected than in fully connected networks, and that
the dominance of only one language is enhanced as the connectivity
decreases. This dominance effect is even stronger in a
two-dimensional space, where domain coarsening tends to drive the
system towards language consensus.