Stochastic population growth and its implications for mutation dynamics under carrying capacity constraints

Mathematical models of mutation processes often assume deterministic population growth, although biological populations are continuously subjected to random fluctuations. Here, we extend a model inspired by the classical Luria-Delbruck framework by introducing stochastic perturbations into population growth while assuming a constant mutation rate. Using Ito calculus, we derive an analytical solution for a stochastic logistic growth model with carrying capacity. By comparing the stochastic model with its deterministic counterpart, we demonstrate that random perturbations can substantially influence population variability, particularly as the population approaches its carrying capacity. While deterministic growth levels off, stochastic fluctuations persist and may become increasingly important relative to the underlying growth dynamics. The magnitude of these effects depends on the relationship between population growth and stochastic volatility. We illustrate the model using real life within-host SARS-CoV-2 infection data and discuss its implications. Our findings highlight the importance of accounting for stochastic population dynamics when modelling mutation-related processes in capacity-limited microbial populations.