Permanent coexistence in chemostat models with delayed feedback control

Feedback control for chemostat model first appears in [P. De Leenher, H.L. Smith, Feedback control for chemostat models, J. Math. Biol. 46 (2003) 48–70] where the authors consider a dilution rate as a feedback control variable, keeping the input nutrient concentration to be constant. They showed that the coexistence of two organisms can be achieved under single nutrient source. In this paper, we improve the model in consideration of a time delay arising from controlling a dilution rate. It is shown that coexistence of two organisms can also be achieved in the improved model. The coexistence is mathematically insisted by solving permanence problem. We prove the problem by using the theory of “average Liapunov function”.