We study the asymmetric nuclear matter in a nonperturbative manner. The bulk nuclear matter studied by the consistent exchange of σ, ω and π mesons is used to investigate its stability. The equation of state (EOS) at zero temperature is considered to study the symmetry energy, curvature parameter of symmetry energy and asymmetry energy. The effect of the density dependence of the symmetry energy on instability property is investigated and analyzed using proton fraction in the nuclear matter. Here a microscopic density-dependent model of the nucleon-meson coupling is used to reexamine the instability of asymmetric nuclear matter.