The heavy-ion fusion reactions induced by neutron-rich nuclei are investigated with the improved quantum molecular dynamics (ImQMD) model. With a subtle consideration of the neutron skin thickness of nuclei and the symmetry potential, the stability of nuclei and the fusion excitation functions of heavy-ion fusion reactions $^{16}$O+$^{76}$Ge, $^{16}$O+$^{154}$Sm, $^{40}$Ca+$^{96}$Zr and $^{132}$Sn+$^{40}$Ca are systematically studied. The fusion cross sections of these reactions at energies around the Coulomb barrier can be well reproduced by using the ImQMD model. The corresponding slope parameter of the symmetry energy adopted in the calculations is $L approx 78$ MeV and the surface energy coefficient is $g_{rm sur}=18pm 1.5$ MeVfm$^2$. In addition, it is found that the surface-symmetry term significantly influences the fusion cross sections of neutron-rich fusion systems. For sub-barrier fusion, the dynamical fluctuations in the densities of the reaction partners and the enhanced surface diffuseness at neck side result in the lowering of the fusion barrier.
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