On stability and stabilization for chemostats with many limiting nutrients

We study models of chemostats where the species compete for multiple limiting nutrients. We first consider cases where the input nutrient concentrations, nutrient flow, and species removal rates are all given positive constants. For such cases, we use Brouwer degree theory to find conditions guaranteeing that the models admit globally asymptotically stable componentwise positive equilibria, starting from all componentwise positive initial states. Then we use our results to prove stabilization results for controlled chemostats with two or more limiting nutrients. When the dilution rate and input nutrient concentrations can be taken as controls, we show that many possible componentwise positive equilibrium points can be rendered globally asymptotically stable. This extends existing control results for chemostats with one limiting nutrient. We illustrate our methods in simulations.