Symmetries and solutions of the non-autonomous von Bertalanffy equation

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. The importance and impact of a closed environment arise when the question under examination relates, for example, to the number 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. Autonomous ODE models are inappropriate as they assume that only the current size of the population controls the growth-decay dynamics. This is reflected in the fact that, asymptotically, their solutions can only grow or decay monotonically or asymptote. Non-autonomous ODE models are not so constrained. A natural strategy for the choice of non-autonomous ODEs is to take appropriate autonomous ones and change them to be non-autonomous through the introduction of relevant non-autonomous terms. This is the approach in this paper with the focus being the von Bertalanffy equation. Since this equation has independent importance in relation to practical applications in growth modelling, it is natural to explore the deeper relationships between the introduced non-autonomous terms through a symmetry analysis, which is the purpose and goal of the current paper. Infinitesimals are derived which allow particular forms of the non-autonomous von Bertalanffy equation to be transformed into autonomous forms for which some new analytic solutions have been found.