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In this paper, we consider a normal branch of the DGP cosmological model with a quintessence scalar field on the brane as the dark energy component. Using the dynamical system approach, we study the stability properties of the model. We find that λ, as one of our new dimensionless variables which is defined in terms of the quintessence potential, has a crucial role in the history of the universe. We divide our discussion into two parts: a constant λ and a varying λ. In the case of a constant λ, we calculate all the critical points of the model even those at infinity and then assume all of them as instantaneous critical points in the varying λ situation which is the main part of this paper. We find that the effect of the extra dimension in such a model is independent of the value of λ. Then, we consider a Gaussian potential for which λ is not constant but varies from zero to infinity. We discuss the evolution of the dynamical variables of the model and conclude that their asymptotic behaviors follow the trajectories of the moving critical points. Also, we find two different possible fates for the universe. In one of them, it could experience an accelerated expansion, but then enters a decelerating phase and finally reaches a stable matter-dominated solution. In the other scenario, the universe could approach the matter-dominated critical point without experiencing any accelerated expansion. We argue that the first scenario is more compatible with observations.