The aim of this thesis is to search for Turing instabilities in Abrams-Strogatz language competition model. The theoretical part of this thesis provides an overview of complex systems, explains the nature of Turing patterns and the conditions for pattern formation, describes the aspects of language dynamics and Abrams-Strogatz model. In the practical part of the thesis, the original Abrams-Strogatz model is extended with spatial parameter and using stability analysis Turing instabilities are investigated in the spatial model