A delayed chemostat model with general nonmonotone response functions and differential removal rates

• Published in: Journal of mathematical analysis and applications Vol. 321; no. 1; pp. 452 - 468
• Main Authors: Wang, Lin, Wolkowicz, Gail S.K.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.09.2006; Elsevier
• Subjects: Biological and medical sciences; Bistability; Chemostat; Competition; Delay differential equations; Exact sciences and technology; Fundamental and applied biological sciences. Psychology; General aspects; Global analysis, analysis on manifolds; Global asymptotic stability; Mathematical analysis; Mathematics; Mathematics in biology. Statistical analysis. Models. Metrology. Data processing in biology (general aspects); Nonmonotone response functions; Ordinary differential equations; Sciences and techniques of general use; Species specific death rates; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Transient dynamics; Competition; Delay differential equations; Global asymptotic stability; Nonmonotone response functions; Species specific death rates; Chemostat; Bistability; Transient dynamics; Chaos; Differential equation; Soliton; Fractals; Delay equation; Convergence; Transients; Asymptotic stability; Survival function; Mathematical analysis; Oscillation; Initial condition; Equilibrium; Response function; Species specific death rate; Sufficient condition; Nutrient
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2005.08.014

A chemostat model with general nonmonotone response functions is considered. The nutrient conversion process involves time delay. We show that under certain conditions, when n species compete in the chemostat for a single resource that is allowed to be inhibitory at high concentrations, the competitive exclusion principle holds. In the case of insignificant death rates, the result concerning the attractivity of the single species survival equilibrium already appears in the literature several times (see [H.M. El-Owaidy, M. Ismail, Asymptotic behavior of the chemostat model with delayed response in growth, Chaos Solitons Fractals 13 (2002) 787–795; H.M. El-Owaidy, A.A. Moniem, Asymptotic behavior of a chemostat model with delayed response growth, Appl. Math. Comput. 147 (2004) 147–161; S. Yuan, M. Han, Z. Ma, Competition in the chemostat: convergence of a model with delayed response in growth, Chaos Solitons Fractals 17 (2003) 659–667]). However, the proofs are all incorrect. In this paper, we provide a correct proof that also applies in the case of differential death rates. In addition, we provide a local stability analysis that includes sufficient conditions for the bistability of the single species survival equilibrium and the washout equilibrium, thus showing the outcome can be initial condition dependent. Moreover, we show that when the species specific death rates are included, damped oscillations may occur even when there is no delay. Thus, the species specific death rates might also account for the damped oscillations in transient behavior observed in experiments.