Stability analysis of a distributed parameter model for the growth of micro-organisms
The principal goal of the paper is to consider a chemostat model where the dynamics are governed in part by a McKendrick-type partial differential equation, and to develop conditions under which the solutions approach the solutions of a system governed by simpler ordinary differential equations. The model is one where the cells in the chemostat have different growth rates due to varying internal concentrations of the nutrient, thus leading to a density function, and hence a first-order partial differential equation describing growth. This is contrasted with the ordinary differential equation describing the growth of the total cell mass assuming cells to be undifferentiated. The author gives the mathematical conditions needed to justify, in the limiting sense, the use of the simpler model, and when that model will have asymptotically stable equilibria. He also considers a more complicated case where the nutrient takes on two forms (e.g., inorganic and organic) and the growth equations depend on the concentrations and rates of change between these two forms. Results of a similar type as in the first case are obtained. The paper is very mathematical, with no examples given, and the conditions needed to be satisfied would be very difficult to be verified in any practical experiment, especially in the second case. Nevertheless, the paper should be regarded as a valuable contribution to the mathematical literature on chemostat models.
McKendrick-type partial differential equation
scientific article; zbMATH DE number 3860941