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 possibly continuous spectrum of phenotypes arising in response to two broad classes of environmental fluctuations that drive changes in the phenotype-dependent growth rates. Specifically, we develop mathematical theory to investigate diversification strategies in environments driven directly by white-noise processes, and in those that manifest growth rates that are continuous almost always. Our primary goal is to demonstrate the existence of regimes in which both discrete and continuous models of heterogeneity are evolutionary advantageous. For tractability, we restrict our analysis to a simplistic exponential growth model, and consider several biologically relevant simplifications that pertain to the relative timescale of phenotype switching. These assumptions yield a series of analytical and semi-analytical expressions that reveal regimes in which heterogeneity is evolutionary advantegeous.