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Gene-mating dynamic evolution theory II: global stability of N-gender-mating polyploid systems.

Juven C Wang1,2,3,4,5

  • 1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA. juven@ias.edu.

Theory in Biosciences = Theorie in Den Biowissenschaften
|February 15, 2020
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Summary

This study generalizes gene-mating models to N genders, finding stable analytic solutions for N-gender, N-polyploid systems with multiple alleles. The research confirms nature supports stable N-gender gene-mating systems without chaos.

Keywords:
Blood types and biological physicsChaotic dynamicsExactly solvable modelsHardy-Weinberg manifoldPopulation genetics and evolutionary biologyTime-dependent nonlinear differential equations

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Area of Science:

  • Population genetics
  • Mathematical biology
  • Evolutionary dynamics

Background:

  • Previous models focused on 2-gender dioecious diploid systems.
  • Gene-mating models describe trait inheritance through allele combinations.
  • Understanding stability in generalized systems is crucial for evolutionary theory.

Purpose of the Study:

  • To determine if Hardy-Weinberg stability and exact analytic solutions exist for N-gender, N-polyploid gene-mating systems.
  • To generalize gene-mating models beyond the traditional 2-gender framework.
  • To investigate the dynamics of systems with an arbitrary number of alleles.

Main Methods:

  • Developed an N-gender N-polyploid gene-mating model.
  • Solved highly nonlinear coupled differential equations governing genotype frequencies.
  • Utilized an analogy to N-body collision Boltzmann equations for mathematical framework.

Main Results:

  • Derived exact analytic solutions for N-gender mating systems with (n+1) alleles for any positive integers N and n.
  • Identified a globally stable solution represented as a continuous manifold.
  • Found no evidence of chaotic behavior in the generalized system.

Conclusions:

  • The Laws of Nature, under the study's assumptions, permit stable N-gender gene-mating systems.
  • The generalized model demonstrates stability and lacks chaotic dynamics.
  • This work provides a theoretical foundation for understanding complex mating systems.