Global stability in chemostat-type equations with distributed delays
Xue-Zhong He, Shigui Ruan and Huaxing Xia
Research Report 96-34
Date: October 1996
We consider a chemostat-type model in which a single
species feeding 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 death of biomass by bacterial
decomposition and the second delay indicates that the growth of the
species depends on the past concentration of 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.
Chemostat-type equations. distributed delay.
Liapunov functionals. local and global stability. nutrient recycling.
AMS Subject Classification (1991)
Primary: 34K15, 34K20, 45J05, 92A15
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Sydney Mathematics and Statistics