The Time Invariance and Nondecreasing Expectation of an Evolutionary Path Characteristic under Weak Selection

Fisher’s fundamental theorem states that the rate of change in mean fitness due to natural selection equals the additive genetic variance in fitness, suggesting that selection generally drives populations toward higher average fitness. Yet in real populations, this increase can be altered or reversed by mutation, frequency-dependent selection, or environmental fluctuations, underscoring the need for complementary measures of adaptation. We analyze stochastic game dynamics of phenotypic frequency in large finite populations under weak frequency-dependent selection and genetic drift. Using the Fokker-Planck equation for the probability density of phenotypic frequency and a path integral formulation, we characterize the feature of possible evolutionary paths over finite timescales. Within this framework, we define the “evolutionary path characteristic” as the likelihood ratio between the probability densities of paths reaching a given endpoint under selection plus drift versus under neutrality. This ratio is time invariant and captures the cumulative effect of directional selection relative to drift. Importantly, its expected value, which equals 1 plus the χ² divergence between evolutionary paths under selection and drift versus those under drift alone, is nondecreasing. In the presence of fitness variance, the effect of directional selection on phenotypic frequency accumulates over time, resulting in a divergence from paths shaped solely by drift. This framework complements Fisher’s theorem by providing a robust measure of accumulating adaptation in stochastic evolutionary dynamics.