Global stability in chemostat-type equations with distributed delays

Xue Zhong He, Shigui Ruan, Huaxing Xia

Research output: Contribution to journalArticle

33 Citations (Scopus)

Abstract

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 (US)
Pages (from-to)681-696
Number of pages16
JournalSIAM Journal on Mathematical Analysis
Volume29
Issue number3
StatePublished - May 1998
Externally publishedYes

Fingerprint

Chemostats
Chemostat
Distributed Delay
Global Stability
Nutrients
Local Stability
Infinite Delay
Biomass
Rate Constant
Physical Model
Model
Limiting
Decompose
Sufficient Conditions
Decomposition
Estimate

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis
  • Applied Mathematics

Cite this

Global stability in chemostat-type equations with distributed delays. / He, Xue Zhong; Ruan, Shigui; Xia, Huaxing.

In: SIAM Journal on Mathematical Analysis, Vol. 29, No. 3, 05.1998, p. 681-696.

Research output: Contribution to journalArticle

@article{bb04ca5b5ca141aba06129a69c36c4b2,
title = "Global stability in chemostat-type equations with distributed delays",
abstract = "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.",
keywords = "Chemostat-type equations, Distributed delay, Liapunov functionals, Local and global stability, Nutrient recycling",
author = "He, {Xue Zhong} and Shigui Ruan and Huaxing Xia",
year = "1998",
month = "5",
language = "English (US)",
volume = "29",
pages = "681--696",
journal = "SIAM Journal on Mathematical Analysis",
issn = "0036-1410",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "3",

}

TY - JOUR

T1 - Global stability in chemostat-type equations with distributed delays

AU - He, Xue Zhong

AU - Ruan, Shigui

AU - Xia, Huaxing

PY - 1998/5

Y1 - 1998/5

N2 - 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.

AB - 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.

KW - Chemostat-type equations

KW - Distributed delay

KW - Liapunov functionals

KW - Local and global stability

KW - Nutrient recycling

UR - http://www.scopus.com/inward/record.url?scp=0032399698&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032399698&partnerID=8YFLogxK

M3 - Article

VL - 29

SP - 681

EP - 696

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 3

ER -