We investigate the properties of the solutions of a class of piecewise-linear differential equations. The equations are appropriate to model biological systems (e.g. genetic networks) in which there are switch-like interactions between the elements. The analysis uses the concept of Filippov solutions of differential equations with a discontinuous right-hand side. It gives an insight into the so-called singular solutions which lie on the surfaces of discontinuity. We show that this notion clarifies the study of several examples studied in the literature.

Although mathematical modelling has reached a degree of maturity in the last decades, microbial ecology is still developing, albeit at a rapid pace thanks to new insights provided by modern molecular tools. However, whilst microbiologists have long enjoyed the perspectives that particular mathematical frameworks can provide, there remains a reluctance to fully embrace the potential of models, which appear too complex, esoteric or distant from the “real-world”. Nevertheless there is a strong case for pursuing the development of mathematical models to describe microbial behaviour and interactions, dynamically, spatially and across scales. Here we put forward perspectives on the current state of mathematical modelling in microbial ecology, looking back at the developments that have defined the synergies between the disciplines, and outline some of the existing challenges that motivate us to provide practical models in the hope that greater engagement with empiricists and practitioners in the microbiological domain may be achieved. We also indicate recent advances in modelling that have had impact in both the fundamental understanding of microbial ecology and its practical application in engineered biological systems. In this way, it is anticipated that interest can be garnered from across the microbiological spectrum resulting in a broader uptake of mathematical concepts in lecture theatres, laboratories and industrial systems