Gene-Mating Dynamic Evolution Theory I: Fundamental assumptions, exactly solvable models and analytic solutions

TL;DR

This paper presents an exact analytic solution to gene-mating evolutionary models, demonstrating stable fixed points and Hardy-Weinberg equilibrium in a broad class of population genetics systems.

Contribution

It introduces a fully solvable mathematical model for gene-mating dynamics with explicit solutions and stability analysis, extending understanding of genetic evolution.

Findings

Genotype frequencies approach a stable fixed point over time.

Alleles that appear in the population never go extinct.

The Hardy-Weinberg law is derived as a natural consequence.

Abstract

Fundamental properties of macroscopic gene-mating dynamic evolutionary systems are investigated. We focus on a single locus, any number of alleles in a two-gender dioecious population, for a large class of systems within population genetics. Our governing equations are time-dependent differential equations labeled by a set of genotype frequencies. Our equations are uniquely derived from 4 assumptions within any population: (1) a closed system; (2) average-and-random mating process (mean-field behavior); (3) Mendelian inheritance; (4) exponential growth/death. Although our equations are nonlinear with time-evolutionary dynamics, we have obtained an exact analytic time-dependent solution and an exactly solvable model. From the phenomenological viewpoint, any initial parameter of genotype frequencies of a closed system will eventually approach a stable fixed point. Under time evolution, we show (1) the monotonic behavior of genotype frequencies, (2) any genotype or allele that appears in the population will never become extinct, (3) the Hardy-Weinberg law, and (4) the global stability without chaos in the parameter space. To demonstrate the experimental evidence for our theory, as an example, we show a mapping from the data of blood type genotype frequencies of world ethnic groups to our stable fixed-point solutions. This fixed-point Hardy-Weinberg manifold, attracting any initial point in any Euclidean fiber bounded within the genotype frequency space to the fixed point where this fiber is attached. The stable base manifold and its attached fibers form a fiber bundle, which fills in the whole genotype frequency space completely. We can define the genetic distance of two populations as their geodesic distance on the equilibrium manifold. The modification of our theory under the process of natural selection and mutation is addressed.