Global Dynamics of a Mathematical Model of Competition in the Chemostat: General Response Functions and Differential Death Rates

A model of exploitative competition of n species in a chemostat for a single, essential, nonreproducing, growth-limiting resource is considered. S. B. Hsu [SIAM J. Appl. Math., 34 (1978), pp. 760763] applies LaSalles extension theorem of Lyapunov stability theory to study the asymptotic behavior of solutions in the special case that the response functions are modeled by MichaelisMenten dynamics. G. J. Butler and G. S. K. Wolkowicz [SIAM J. Appl. Math., 45 (1985), pp. 138151], on the other hand, allow more general response functions (including monotone and nonmonotone functions), but their analysis requires the assumption that the death rates of all the species are negligible in comparison with the washout rate, and hence can be ignored. By means of Lyapunov stability theory, the global dynamics of the model for a large class of response functions are studied, including both monotone and nonmonotone functions (though it is not as general as the class studied by Butler and Wolkowicz) and the results in Hsu are extended for this class, to the differential death-rate case. That is, it is shown that for this class the outcome depends on the relative sizes of the break-even concentrations. Provided that these concentrations are distinct, at most one competitor population avoids extinction, the one with the lowest break-even concentration. All populations approach limiting values.