## Abstract

Extending the previous 2-gender dioecious diploid 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 N-polyploid gene-mating system with 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.

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## Notes

- 1.
For example, in Wang and Chen (2020), the (\(n+1\))-alleles can be regarded as

*n*dominant alleles and 1 recessive allele in a single locus. We can denote the 1 recessive allele as \({\mathcal {G}}_0\), and the*n*dominant alleles as \({\mathcal {G}}_1, \ldots , {\mathcal {G}}_n\). - 2.
For the sake of keeping the minimal amount of notations, later in all sections, we will map the genotype population \({G}_{\dots }\) to genotype frequency \({G'_{\dots }}\), \({G'_{\dots }} \equiv \frac{{G}_{\dots }}{P}\), then rename the genotype frequency as \({G_{\dots }}\).

- 3.
Here the genotype label \({ \underset{k_{0}}{\underbrace{\alpha _{0}\cdots \alpha _{0}}} \underset{k_{1}}{\underbrace{\alpha _{1}\cdots \alpha _{1}}} \dots \dots \underset{ k_{m}}{\underbrace{\alpha _{m}\cdots \alpha _{m}}}}\) in the continuous genotype frequency distribution function \({G_{ \underset{k_{0}}{\underbrace{\alpha _{0}\cdots \alpha _{0}}} \underset{k_{1}}{\underbrace{\alpha _{1}\cdots \alpha _{1}}} \dots \dots \underset{ k_{m}}{\underbrace{\alpha _{m}\cdots \alpha _{m}}}}}(t)\) is a discretized labeling, while the time

*t*in our model is continuous. - 4.
In contrast, the conventional Boltzmann equation has the continuous variables \((\vec {x}, \vec {p})\) in the continuous distribution function \(f(\vec {x}, \vec {p},t)\), e.g., \((\vec {x} \in {\mathbb {R}}, \vec {p} \in {\mathbb {R}})\).

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## Acknowledgements

JW acknowledges the NSF Grant PHY-1606531 and the support from Institute for Advanced Study. This work is also supported by NSF Grant DMS-1607871 “Analysis, Geometry and Mathematical Physics” and Center for Mathematical Sciences and Applications at Harvard University. This work is supported by NSF Grant Nos. DMR-1005541 and NSFC 11274192. It is also supported by the BMO Financial Group and the John Templeton Foundation. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research. This work was supported in part by the MOST, NTUCTS, the NTU-CASTS of R.O.C, by the Ministry of Science and Technology, Taiwan, under Grant No. 108-2112-M-002-003-MY3 and the Kenda Foundation.

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Wang, J.C. Gene-mating dynamic evolution theory II: global stability of N-gender-mating polyploid systems.
*Theory Biosci.* (2020). https://doi.org/10.1007/s12064-020-00308-4

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### Keywords

- Population genetics and evolutionary biology
- Chaotic dynamics
- Blood types and biological physics
- Exactly solvable models
- Time-dependent nonlinear differential equations
- Hardy-Weinberg manifold