Mathematical model of closed microecosystem “algae – heterotrophic bacteria”

Model of closed microecosystem “algae – heterotrophic bacteria” is proposed in this paper. Mathematical model is the Cauchy problem for system of nonlinear ordinary differential equations. To develop the model the Liebig’s law of the minimum is consistently used for both specific rate of biomass growth and specific death rate of algae and bacteria cells. To describe the specific rate of substrate utilization by algae and bacteria the Andrew’s model (substrate inhibition) is used. It is assumed that specific death rate of algae and bacteria cells increases with decreasing substrate concentration. It is also assumed that carbon and nitrogen are main biogenic elements, and in the system they are in the form of mineral substrate (CO$_2$, NO$_2$, NO$_3$, NH$_4$) and biological substrate (proteins, lipids and carbohydrates). Mathematical model describing time variations in concentration of elements of microecosystem is formulated under the following assumptions: 1) stoichiometric coefficients of algae and bacteria cells are constant in the development of microecosystem; 2) utilization of carbon and nitrogen by algae and bacteria occurs independently; 3) oxygen produced by algae cells during photosynthesis completely covers the demand for oxygen for algae and bacteria cells. To verify the proposed model experimental data for microecosystems “Clorella vulgaris – Pseudomonas sp.” и “Scenedesmus obliquus – Pseudomonas sp.” are used. These systems were studied in laboratory conditions, and concentrations of elements of microecosystems in stationary state were obtained. Parameters of functions describing specific rate of utilization of biogenic elements were derived from experimental data for growth kinetics of algae and bacteria. Concentration of the biomass in stationary state obtained with the use of the proposed model is in reasonable agreement with experimental data.