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Global periodic solutions for a differential delay system modeling a microbial population in the chemostat. (English) Zbl 0833.34069
The author studies the differential delay system modeling a microbial population in the chemostat $$\dot s(t)= 1- s(t)- P(s(t))u(t),\quad \dot u(t)= [- 1+ P(S((t- \tau))].\tag1$$ The author gives a sufficient conditions that the washout steady state ($u= 0$, $S= 1$) is globally stable and establishes sufficient conditions for the global existence of a periodic solution by proving the existence of nontrivial periodic points of an appropriate map. The author also presents an explicit application to Michaelis-Menten kinetics.

##### MSC:
 34K13 Periodic solutions of functional differential equations 92D25 Population dynamics (general) 34C25 Periodic solutions of ODE
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