Phenotypic heterogeneity in temporally fluctuating environments

Many biological systems regulate phenotypic heterogeneity as a fitness-maximising strategy in uncertain and dynamic environments. Analysis of such strategies is typically confined both to a discrete set of environmental conditions, and to a discrete (often binary) set of phenotypes specialised to each condition. In this work, we extend theory on both fronts to encapsulate a potentially continuous spectrum of phenotypes arising in response to environmental fluctuations that drive changes in the phenotype-dependent growth rate. We consider two broad classes of stochastic environment: those that are temporally uncorrelated (modelled by white-noise processes), and those that are correlated (modelled by Poisson and Ornstein-Uhlenbeck processes). For tractability, we restrict analysis to an exponential growth model, and consider biologically relevant simplifications that pertain to the timescale of phenotype switching relative to fluctuations in the environment. These assumptions yield a series of analytical and semi-analytical expressions that reveal environments in which phenotypic heterogeneity is evolutionarily advantageous.