A topic for integrated teaching of mathematics and biology: the parabola of chaos in tumour cell aneuploidy

Biological systems exhibit strong dynamic variations; for example, a system may have a tendency towards either a steady state, periodic behaviour or chaotic oscillations. These findings, explained by dynamic systems theory, have allowed researchers to formally relate a wide variety of systems, such as the theory of rumour propagation, glucose metabolism or the activity of a neuronal circuit. Interestingly, among all of the dynamic characteristics mentioned, qualitative aspects frequently observed in highly complex systems can be obtained through the recursive application of a very simple parabolic function known as the logistic equation. Currently, the importance of these dynamic phenomena for biomedicine is recognized, but simple models such as the logistic equation have not been sufficiently exploited in the training of biologists and health professionals. We briefly review some properties of the dynamics governed by the logistic equation, with a direct application to the theory of cancer based on the dynamics of aneuploidy. We believe that this already known mathematical application is an opportunity for integrated teaching of mathematics and biology. For this purpose we finally provide an appendix with some activities, all of them related to this article, which can be practiced in a simple spreadsheet that we have shared.