# Stochastic and Information-thermodynamic Structures of Population Dynamics in Fluctuating Environment

## TL;DR

This paper explores the fundamental stochastic and thermodynamic principles underlying how living systems adapt to fluctuating environments, revealing limits of fitness gain and conditions for optimal adaptation through derived fluctuation relations.

## Contribution

It introduces causal fluctuation relations linking fitness and information, generalizes the evolutionary stable state concept, and clarifies the thermodynamic limits of adaptation in fluctuating environments.

## Key findings

- Derived causal fluctuation relations of fitness and information.
- Identified the limit of fitness gain and conditions for excess fitness.
- Generalized the concept of evolutionary stable state for fluctuating environments.

## Abstract

Adaptation in a fluctuating environment is a process of fueling environmental information to gain fitness. Living systems have gradually developed strategies for adaptation from random and passive diversification of the phenotype to more proactive decision making, in which environmental information is sensed and exploited more actively and effectively. Understanding the fundamental relation between fitness and information is therefore crucial to clarify the limits and universal properties of adaptation. In this work, we elucidate the underlying stochastic and information-thermodynamic structure in this process, by deriving causal fluctuation relations (FRs) of fitness and information. Combined with a duality between phenotypic and environmental dynamics, the FRs reveal the limit of fitness gain, the relation of time reversibility with the achievability of the limit, and the possibility and condition for gaining excess fitness due to environmental fluctuation. The loss of fitness due to causal constraints and the limited capacity of real organisms is shown to be the difference between time-forward and time-backward path probabilities of phenotypic and environmental dynamics. Furthermore, the FRs generalize the concept of evolutionary stable state (ESS) for fluctuating environment by giving the probability that the optimal strategy on average can be invaded by a suboptimal one owing to rare environmental fluctuation. These results clarify the information thermodynamic structures in adaptation and evolution.