Environmental stochastic fluctuations lead to different evolutionary directions of strategies

Though natural selection posits that biological evolution is directional to increase individual fitness, in real populations the evolutionary process might be greatly affected by stochastic factors such as environmental fluctuations. These stochastic factors introduce directional uncertainty, a phenomenon that remains underexplored. To explore how environmental stochastic fluctuations influence the direction of evolution, we analyze the dynamics of a two-strategy evolutionary game with a stochastic payoff matrix in a fixed finite-size population under weak selection. Within the path integral framework, we derive the transition probability density p(x,t;x₀) of strategy frequencies in finite time and identify the evolutionary characteristic γ(x,x₀)=p(x,t;x₀)/p̃(x,t;x₀) that quantifies the evolutionary direction, where p̃(x,t;x₀) corresponds to the density under neutrality. Results show that under weak selection, when low-intensity environmental fluctuations (i.e., weak payoff noise) occur, the evolutionary direction depends on the mean values and is independent of the variances of stochastic payoffs. This direction exhibits consistency with a deterministic environment without stochastic fluctuations. However, when the payoff noise is strong, the evolutionary direction is jointly determined by both the mean values and variances of the payoffs, and it will alter the evolutionary direction of a deterministic environment. Furthermore, based on evolutionary characteristics, results from the stochastic Prisoner's Dilemma suggest that natural selection favors cooperation when the variances of defection payoffs increase.