Microbial growth in periodic gradostats. (English) Zbl 0726.92025
A gradostat is a laboratory device with which one can study the growth of microorganisms in a nutrient gradient. As constructed by R. W. Lovitt and J. W. T. Wimpenny [Soc. Gen. Microbial Quart. 6 (1979)] a gradostat is a concatenation of several chemostats in which adjacent vessels are connected by tubes allowing pumps to exchange the material contents of each vessel. The aim of the paper is to study a mathematical model of the growth of a single species of microorganisms in the presence of one limiting substrate or two limiting complementary substrates in a very general gradostat where essentially arbitrary connections between vessels are allowed. In addition, we allow operating parameters such as flow rates and reservoir nutrient concentration to vary periodically in time, simulating seasonal or diurnal variations in natural environments.
The mathematical model reduces to a system of periodic, nonautonomous quasimonotone ordinary differential equations for species concentration in each vessel. The extinction or persistence of the population in the gradostat is determined by whether a certain dominant Floquet multiplier of a known quasimonotone linear periodic system either does not or does exceed unity. If the multiplier exceeds one then all nontrivial solutions are shown to approach a unique periodic steady state oscillation.
The mathematical model reduces to a system of periodic, nonautonomous quasimonotone ordinary differential equations for species concentration in each vessel. The extinction or persistence of the population in the gradostat is determined by whether a certain dominant Floquet multiplier of a known quasimonotone linear periodic system either does not or does exceed unity. If the multiplier exceeds one then all nontrivial solutions are shown to approach a unique periodic steady state oscillation.
Reviewer: Hal Smith
MSC:
| 92D40 | Ecology |
| 92C99 | Physiological, cellular and medical topics |
| 34C25 | Periodic solutions to ordinary differential equations |
Keywords:
seasonal variations; Perron-Frobenius theory; monotone dynamics; cooperative systems; gradostat; growth of microorganisms; chemostats; limiting substrate; limiting complementary substrates; flow rates; reservoir nutrient concentration; diurnal variations; system of periodic, nonautonomous quasimonotone ordinary differential equations; extinction; persistence; Floquet multiplier; periodic steady state oscillationReferences:
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