  We have studied homogeneous isotropic FRW model having dynamical dark energy
DBI-essence with scalar field. The existence of cosmological scaling solutions
restricts the Lagrangian of the scalar field $\phi$. Choosing $p=X g(X
e^{\lambda \phi})$, where $X=-g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi
/2$ with $g$ is any function of $X e^{\lambda \phi}$ and defining some suitable
transformations, we have constructed the dynamical system in different gravity:
(i) Loop Quantum Cosmology (LQC), (ii) DGP BraneWorld and (iii) RS-II Brane
World. We have investigated the stability of this dynamical system around the
critical point for three gravity models and investigated the scalar field
dominated attractor solution in support of accelerated universe. The role of
physical parameters have also been shown graphically during accelerating phase
of the universe.
