Exponential growth is one of the first mathematical models we are introduced to in a differential questions course, yet many seemingly basic questions about the phenomenology of exponential growth are non-trivial. For example, in a population of cells undergoing binary fission with random generation times, how is the distribution of generation times related to the long-term exponential growth rate of the population? Similar questions were already being asked by Euler in the 18th century, but thanks to recent experimental innovations which enable researchers to record individual cells for long periods of time in highly controlled conditions, an entire new set of questions about exponential growth, especially stochastic exponential growth, have emerged.
In this talk, I will discuss some classical results from Euler and Lotka before I present two examples from my own research. First, I will discuss the question of whether the dynamics of a microbial population (in particular, it’s rate of growth) can be predicted from measurements of single-cells. I will derive an algorithm for linking single-cell dynamics to population growth, which reveals a connection to Large deviation theory. Second, I discuss the question: what makes genetically identical cells grow at different rates? In both cases, we will see how biological data can inspire new mathematical questions and analysis techniques.