Gene-Mating Dynamic Evolution Theory II: Global stability of N-gender-mating polyploid systems

TL;DR

This paper generalizes gene-mating evolution models to N-gender systems, providing exact solutions and proving global stability without chaos, extending previous 2-gender models to more complex polyploid systems.

Contribution

It offers the first exact analytic solutions for N-gender N-polyploid gene-mating systems with arbitrary alleles, demonstrating their global stability and absence of chaos.

Findings

Exact solutions for N-gender gene-mating equations

Global stability of the N-gender systems

No chaotic behavior observed in the models

Abstract

Extending the previous 2-gender dioecious diploid gene-mating evolution model [arXiv:1410.3456], we attempt to answer "whether the Hardy-Weinberg global stability and the exact analytic dynamical solutions can be found in the generalized N-gender N-polyploid gene-mating system with an arbitrary number of alleles?" For a 2-gender gene-mating evolution model, a pair of male and female determines the trait of their offspring. Each of the pair contributes one inherited character, the allele, to combine into the genotype of their offspring. Hence, for an N-gender N-polypoid gene-mating model, each of N different genders contributes one allele to combine into the genotype of their offspring. We exactly solve the analytic solution of N-gender-mating $(n+1)$-alleles governing highly-nonlinear coupled differential equations in the genotype frequency parameter space for any positive integer N and $n$. For an analogy, the 2-gender to N-gender gene-mating equation generalization is analogs to the 2-body collision to the N-body collision Boltzmann equations with continuous distribution functions of "discretized" variables instead of "continuous" variables. We find their globally stable solution as a continuous manifold and find no chaos. Our solution implies that the Laws of Nature, under our assumptions, provide no obstruction and no chaos to support an N-gender gene-mating stable system.