## 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 $\dot s(t)= 1- s(t)- P(s(t))u(t),\quad \dot u(t)= [- 1+ P(S((t- \tau))].\tag{1}$ 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.
Reviewer: Li Jibin (Kunming)

### MSC:

 34K13 Periodic solutions to functional-differential equations 92D25 Population dynamics (general) 34C25 Periodic solutions to ordinary differential equations
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