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

Fundamental properties of macroscopic gene-mating dynamic evolutionary systems are investigated. A model is proposed to describe a large class of systems within population genetics. We focus on a single locus, arbitrary number alleles in a two-gender dioecious population. Our governing equations are time-dependent continuous differential equations labeled by a set of genotype frequencies. The full parameter space consists of all allowed genotype frequencies. Our equations are uniquely derived from four fundamental assumptions within any population: (1) a closed system; (2) average-and-random mating process (mean-field behavior); (3) Mendelian inheritance; (4) exponential growth and exponential death. Even though our equations are non-linear with time evolutionary dynamics, we have an exactly solvable model. Our findings are summarized from phenomenological and mathematical viewpoints. From the phenomenological viewpoint, any initial genotype frequency 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, as an example, we show a mapping from the blood type genotype frequencies of world ethnic groups to our stable fixed-point solutions. From the mathematical viewpoint, the equilibrium solutions consist of a base manifold as a global stable attractor, attracting any initial point in a Euclidean fiber bundle to the fixed point where the fiber is attached. We can define the genetic distance of two populations as their geodesic distance on the equilibrium manifold. In addition, the modification of our theory under the process of natural selection and mutation is addressed.

Comments: 19 + 11 pages, 15 figures, 6 tables. See Tables and Figures for a summary of key results. v2: figure display format issue fixed, Refs added, and refinement

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