The distribution of human linguistic groups presents a number of interesting and non-trivial patterns. The distributions of the number of speakers per language and the area each group covers follow log-normal distributions, while population and area fulfill an allometric relationship. The topology of networks of spatial contacts between different linguistic groups has been recently characterized, showing atypical properties of the degree distribution and clustering, among others. Human demography, spatial conflicts, and the construction of networks of contacts between linguistic groups are mutually dependent processes. Here we introduce an adaptive network model that takes all of them into account and successfully reproduces, using only four model parameters, not only those features of linguistic groups already described in the literature, but also correlations between demographic and topological properties uncovered in this work. Besides their relevance when modeling and understanding processes related to human biogeography, our adaptive network model admits a number of generalizations that broaden its scope and make it suitable to represent interactions between agents based on population dynamics and competition for space.