A two-species system is considered, in which irreversible aggregations occur between particles of the same species while irreversible complete annihilations occur between particles of different species. Such competing processes between aggregation and annihilation reactions are performed by Monte Carlo simulations under various parameter conditions, and the influences of aggregation rate, annihilation rate, and initial particle distribution on the dynamics of the system are analyzed in detail. Simulation results indicate that the particle size distributions always obey a certain scaling law. When the aggregation rates of the two kinds of particles are both twice as fast as the annihilation rate, the scaling exponents of the particle size distributions have relation with the initial particle distribution; while in the remaining cases, the scaling exponents depend crucially on the reaction rates. Moreover, when both aggregation rates are larger than or equal to the double of the annihilation rate, all particles will annihilate each other completely; while at least one of the aggregation rates is slower than the double of the annihilation rate, the species with slower aggregation rate could survive finally. Simulation results are in good agreement with the reported theoretical solutions.