Discrete-time models with mosquitoes carrying genetically-modified bacteria

We formulate a homogeneous model and a stage-structured model for the interactive wild mosquitoes and mosquitoes carrying genetically-modified bacteria. We establish conditions for the existence and stability of fixed points for both models. We show that a unique positive fixed point exists and is asymptotically stable if the two boundary fixed points are both unstable. The unique positive fixed point exists and is unstable if the two boundary fixed points are both locally asymptotically stable. Using numerical examples, we demonstrate the models undergoing a period-doubling bifurcation.