Gene-mating dynamic evolution theory: fundamental assumptions, exactly solvable models and analytic solutions.

Area of Science:

• Population Genetics
• Evolutionary Systems
• Mathematical Biology

Background:

• Gene-mating dynamics are crucial for understanding population genetics.
• Existing models often lack exact analytic solutions for complex systems.
• Macroscopic properties of evolutionary systems require robust theoretical frameworks.

Purpose of the Study:

• To investigate fundamental properties of macroscopic gene-mating dynamic evolutionary systems.
• To develop an exactly solvable model for a dioecious population with any number of alleles at a single locus.
• To analyze the long-term behavior of genotype frequencies under specific evolutionary assumptions.

Main Methods:

• Developed time-dependent continuous differential equations based on four core assumptions: closed system, random mating, Mendelian inheritance, and exponential growth/death.
• Obtained an exact analytic time-dependent solution for the nonlinear system.
• Analyzed phenomenological and mathematical properties of the solutions, including stability and attractor manifolds.

Main Results:

• Demonstrated that genotype frequencies in a closed system asymptotically approach a stable fixed point.
• Showed monotonic behavior of genotype frequencies, ensuring no allele extinction.
• Confirmed adherence to the Hardy-Weinberg law and global stability without chaos.
• Identified a continuous manifold of stable equilibrium solutions (Hardy-Weinberg manifold).

Conclusions:

• The developed model provides an exactly solvable framework for population genetics.
• The Hardy-Weinberg manifold acts as a global stable attractor for genotype frequencies.
• The theory offers a method to define genetic distance between populations and can be extended to include natural selection and mutation.