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

Extending the previous 2-gender dioecious biploid gene-mating evolution model, we attempt to answer “whether the Hardy-Weinberg global stability and the exact analytic dynamical solutions can be found in the generalized N-gender polyploid gene-mating system?” 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 polypoid gene-mating model, each of N different genders contributes one allele to combine into the genotype of their offspring. We find the global stable solution as a manifold in the genotype frequency parameter space. We exactly sovle the analytic solution of N-gender mating governing equations and find no chaos. The 2-gender to N-gender gene-mating equation is analogues to the 2-body collision to the N-body collision Boltzmann equations with discretized distribution.