Global periodic solutions for a differential delay system modeling a microbial population in the chemostat. (English) Zbl 0833.34069

Zhao, Tao

J. Math. Anal. Appl. 193, No. 1, 329-352 (1995).

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

Keywords:

stability; differential delay system; microbial population in the chemostat; washout steady state; periodic solution; Michaelis-Menten kinetics