All-optical switch and multiple logic gates have been demonstrated using a hybrid-cavity semiconductor laser composed of a square microcavity and a Fabry-Perot cavity experimentally. In this paper, two-section tri-mode rate equations with optical injection terms are proposed and applied to study all-optical logic gates of NOT, NOR, and NAND operations utilizing the hybrid-cavity laser. Steady-state and dynamical characteristics of all-optical multiple logic gates are simulated, taking into account the influence of mode frequency detuning, gain suppression coefficients, mode Q factor, injection energy, and biasing current. All-optical logic NOT, NOR, and NAND gates up to 20, 15, and 20 Gbit/s are obtained numerically with dynamic extinction ratios of over 20, 20, and 10 dB, respectively, which are potential response speeds of the all-optical logic gates based on the hybrid-cavity semiconductor lasers.