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

Area of Science:

• Evolutionary biology
• Population genetics
• Mathematical modeling

Background:

• Fisher's fundamental theorem suggests natural selection increases mean fitness.
• Real populations face factors like mutation and drift that alter fitness trajectories.
• Complementary measures are needed to assess adaptation under complex evolutionary dynamics.

Purpose of the Study:

• To analyze stochastic game dynamics of phenotypic frequency in large finite populations.
• To characterize evolutionary paths over finite timescales under weak frequency-dependent selection and genetic drift.
• To define and evaluate a new measure for accumulating adaptation.

Main Methods:

• Utilized the Fokker-Planck equation to model probability density of phenotypic frequency.
• Employed a path integral formulation to analyze evolutionary trajectories.
• Defined the "evolutionary path characteristic" as a likelihood ratio of path probabilities.

Main Results:

• Introduced the time-invariant "evolutionary path characteristic" to quantify directional selection relative to drift.
• Demonstrated that the expected value of this characteristic is nondecreasing, reflecting accumulated adaptation.
• Showed that fitness variance causes divergence from neutral drift paths.

Conclusions:

• The "evolutionary path characteristic" provides a robust measure of accumulating adaptation in stochastic evolutionary dynamics.
• This framework complements Fisher's theorem by capturing adaptation beyond average fitness changes.
• The study offers new tools for understanding evolutionary trajectories in finite populations.