Global stability in chemostat-type equations with distributed delays

Xue Zhong He*, Shigui Ruan, Huaxing Xia

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

38 Citations (Scopus)


We consider a chemostat-type model in which a single species feeds on a limiting nutrient supplied at a constant rate. The model incorporates a general nutrient uptake function and two distributed (infinite) delays. The first delay models the fact that the nutrient is partially recycled after the death of the biomass by bacterial decomposition, and the second delay indicates that the growth of the species depends on the past concentration of the nutrient. By constructing appropriate Liapunov-like functionals, we obtain sufficient conditions for local and global stability of the positive equilibrium of the model. Quantitative estimates on the size of the delays for local and global stability are also obtained with the help of the Liapunov-like functionals. The technique we use in this paper may be used as well to study global stability of other types of physical models with distributed delays.

Original languageEnglish
Pages (from-to)681-696
Number of pages16
JournalSIAM Journal on Mathematical Analysis
Issue number3
Publication statusPublished - May 1998
Externally publishedYes


  • Chemostat-type equations
  • Distributed delay
  • Liapunov functionals
  • Local and global stability
  • Nutrient recycling


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