Lead is an important alloy material and nuclear fuel component. Lead-based eutectic alloys serve as important coolants and have been extensively utilized in the construction of lead-cooled fast reactor, such as the European lead-cooled System (ELSY) and the China lead-based Research reactor (CLEAR-I). These materials also play a significant role in research related to Generation-IV reactors. The study and calculation of lead nuclear data have important theoretical value and application prospects. ²⁰⁸Pb is the most stable and abundant isotope in lead nuclei, and high-quality description of ²⁰⁸Pb nuclear scattering data is important in achieving accurate theoretical calculations of nuclear reaction cross-sections in lead-based nuclear systems Based on the dispersive optical model, the nucleon scattering on ²⁰⁸Pb is described through the implementation of a dispersive optical potential in this work. The dispersive optical model potential is defined as energy-dependent real potential and imaginary potential. The dispersive contribution corresponding to the real potential is calculated analytically from the corresponding imaginary potential by using a dispersion relation, and the isospin dependence is reasonably considered by introducing an isovector component (i.e. Lane term) into the real part and the imaginary part of potential: the depth constant of the real Hartree-Fock potential $ V_{\rm{HF}}$ and the depth constant of surface imaginary potential $ W_{\rm{s}}$. Unlike K-D potential, which requires two different sets of parameters to describe neutron and proton induced scattering data. This optical potential uses the same set of parameters to simultaneously describe nucleon-nucleus scattering data. The derived potential in this work shows a very good description of nucleon-nucleus scattering data on ²⁰⁸Pb with energies up to 200 MeV. The calculated neutron total cross sections, neutron and proton elastic scattering angular distributions, and neutron and proton elastic analyzing powers are shown to be in good agreement with experimental data. Additionally, the difference in potential between neutrons and protons induced is described by an isovector term, achieving the reasonable and good prediction of quasielastic (p, n) scattering data.