## JMI2013A-5 Modelling microbial growth in a closed environment (pp.33-40)

Author(s)： Maureen P. Edwards, Ulrike Schumann and Robert S. Anderssen

J. Math-for-Ind. **5A** (2013) 33-40.

- File： JMI2013A-5.pdf (142KB)

Abstract

The formulation of models for the growth of microbes (fungi; bacteria) must not only take account of the current number of the microbes but also of the effect of the environment in which the growth is occurring and of the type of measurements used to record the growth. In industrial processes, the effect of physiological (morphological) changes on the growth must be taken into account. When growth is occurring in an open environment which achieves an equilibrium (e.g. balancing growth and harvesting) and is measured as the time evolution of the total number of microbes present (alive and dead), autonomous ordinary differential equation (ODE) models are appropriate. The corresponding growth measurements (optical density; centrifuged weight), because they record the total number of microbes present (alive and dead), have a logistic structure which autonomous equations, such as the Verhulst, capture. For growth in a closed environment, which is indicative of the situation in laboratory experiments, autonomous ODE models do not necessarily capture the dynamics under investigation. Such situations arise when the question under examination relates to the activity of the surviving microbes, such as in a study of the spoilage and contamination of food, the gene silencing activity of fungi or the production of a chemical compound by bacteria or fungi. Practical and theoretical implications associated with the measurement and modelling of the number of surviving microbes in a closed environment is the focus in this paper. The limitations of current measurement protocols to track the number of surviving microbes are discussed. The use of non-autonomous modifications of autonomous ODE models of growth is proposed and analysed. In particular, a non-autonomous version of the von Bertalanffy model is proposed as an appropriate framework in which to analyse growth in a closed and/or deteriorating environment.

Keyword(s). growth modelling, autonomous, non-autonomous, microbial growth, ordinary differential equations, closed environment, food contamination, surviving microbes