How many species can two essential resources support?

A chemostat model of n species of microorganisms competing for two perfectl y complementary, growth-limiting nutrients is considered. Sufficient condit ions are given for there to be a single winning species and for two species to coexist, driving the others to extinction. In the case when n=3, it is shown that every solution converges to one of the single-species or two-spe cies steady states, and hence the dynamics of the model is completely deter mined. The results generalize those of Hsu, Cheng, and Hubbell [SIAM J. App l. Math., 41 (1981), pp. 422-444] as well as Butler and Wolkowicz [Math. Bi osci., 83 (1987), pp. 1-48] who considered two species.