Global stability in chemostat-type equations with distributed delays

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.