Fluctuation Relations of Fitness and Information in Population Dynamics

Phenotype switching with and without sensing environment is a common strategy of organisms to survive in a fluctuating environment. Understanding the evolutionary advantages of switching and sensing requires a quantitative evaluation of their fitness gain and its fluctuation together with the conditions for the switching and sensing strategies being adapted to a given environment. In this work, by using a pathwise formulation of the population dynamics, we show that the optimal switching strategy is characterized by a consistency condition for time-forward and backward path probabilities. The formulation also clarifies the underlying information-theoretic aspect of selection as a passive information compression. The loss of fitness by a suboptimal strategy is also shown to satisfy a fluctuation relation, which provides us with the information on how environmental fluctuation impacts the advantages of the optimal strategy. These results are naturally extended to the situation that organisms can use an environmental signal by actively sensing the environment. The fluctuation relations of the fitness gain by sensing are derived in which the multivariate mutual information among the phenotype, the environment, and the signal plays the role to quantify the relevant information in the signal for the fitness gain.