We extend the effective dynamical quasi-particle model (DQPM) - constructed for the description of non-perturbative QCD phenomena of the strongly interacting quark-gluon plasma (QGP) - to large baryon chemical potentials including a critical end-point (CEP) and a 1st order phase transition. The DQPM is based on covariant propagators for quarks/antiquarks and gluons that have a finite width in their spectral functions. In DQPM the determination of complex selfenergies for the partonic degrees-of-freedom at zero and finite $mu_B$ has been performed by adjusting the entropy density to the lattice QCD data. The temperature-dependent effective coupling (squared) $g^2(T/T_c)$, as well as the effective masses and widths or the partons are based in this adjustment. The novel extended dynamical quasi-particle model, named DQPM-CP, makes it possible to describe thermodynamical and transport properties of quarks and gluons in a wide range of temperature, $T$, and baryon chemical potential, $mu_B$, and reproduces the equation-of-state (EoS) of lattice QCD calculations in the crossover region of finite $T, mu_B$. We apply a scaling ansatz for the strong coupling constant near the CEP, located at ($T^{CEP}$, $mu^{CEP}_B) = (0.100, 0.960)$ GeV. We show the EoS as well as the speed of sound for $T>T_c$ and for a wide range of $mu_B$, which can be of interest for hydrodynamical simulations. Furthermore, we consider two settings for the strange quark chemical potentials (I) $mu_q=mu_u=mu_s=mu_B/3$ and (II) $mu_s=0,mu_u=mu_d=mu_B/3$. The isentropic trajectories of the QGP matter are compared for these two cases. The phase diagram of DQPM-CP is close to PNJL calculations. The leading order pQCD transport coefficients of both approaches differ. This elucidates that the knowledge of the phase diagram alone is not sufficient to describe the dynamical evolution of strongly interacting matter.
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