Exact Computation of Growth-Rate Variance in Randomly Fluctuating Environment

We consider a general class of Markovian models describing the growth in a randomly fluctuating environment of a clonal biological population having several phenotypes related by stochastic switching. Phenotypes differ e.g. by the level of gene expression for a population of bacteria. The time-averaged growth rate of the population, Λ , is self-averaging in the limit of infinite time; it may be understood as the fitness of the population in a context of Darwinian evolution. The observation time T being however typically finite, the growth rate fluctuates. For T finite but large, we obtain the variance of the time-averaged growth rate as the maximum of a functional based on the stationary probability distribution for the phenotypes. This formula is general. In the case of two states, the stationary probability was computed by Hufton et al. (J Stat Mech 2018:23501, 2018), allowing for an explicit expression which can be checked numerically. Applications of our main formula to the study of survival strategies of biological populations, as developed in the companion article (Dinis et al. in http://doi.org/10.1101/2022.01.18.476793v1 ), are presented here briefly.