this work, we investigate a Cournot-Bertrand duopoly model with product differentiation between two different firms, where one company uses price as a decision variable and the other uses quantity. The stability of the Nash equilibrium point is investigated, and the role of the parameters on the model stability is studied. It is shown that the system experiences three different types of bifurcations when the parameters go through the different boundary curves of the stability region. Furthermore, the model's bifurcation is controlled using a hybrid control strategy.