The invariance and non-decreasing expectation of an evolutionary path characteristic under weak selection

Abstract: Fisher's fundamental theorem of natural selection states that the rate of change in a population's mean fitness equals its additive genetic variance in fitness. This implies that mean fitness should not decline in a constant environment, thereby positioning it as an indicator of evolutionary progression. However, this theorem has been shown to lack universality. Here, we derive the Fokker-Planck equation that describes the stochastic frequency dynamics of two phenotypes in a large population under weak selection and genetic drift, and develop a path integral formulation that characterizes the probability density of phenotypic frequency. Our formulation identifies that, under both selection and genetic drift, the ratio of the probability density of adaptive traits (e.g., phenotypic frequency) to that under neutrality represents a time-invariant evolutionary path characteristic. This ratio quantifies the cumulative effect of directional selection on the evolutionary process compared to genetic drift. Importantly, the expected value of this ratio does not decline over time. In the presence of fitness variance, the effect of directional selection on expected phenotypic changes accumulates over time, diverging progressively from paths shaped solely by genetic drift. The expectation of this time-invariant ratio thus offers a robust and informative alternative to mean fitness as a measure of progression in stochastic evolutionary dynamics.