  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 {\lambda}, 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
{\lambda} and a varying {\lambda}. In the case of a constant {\lambda}, 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 {\lambda}
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 {\lambda}. Then,
we consider a Gaussian potential for which {\lambda} 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.
