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.

中文翻译：

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基因交配动态进化理论II：N性别交配多倍体系统的整体稳定性。

在扩展先前的两性两性二倍体基因交配进化模型之前，我们试图回答“是否能在所有具有任意性的广义N性别N多倍体基因交配系统中找到Hardy-Weinberg全局稳定性和精确的解析动力学解。等位基因数量？” 对于2性别的基因交配进化模型，一对雌雄决定其后代的特征。这对中的每一个都贡献一个遗传特征，即等位基因，以结合到其后代的基因型中。因此，对于一个N性别N息肉基因配对模型，N个不同性别中的每一个都贡献一个等位基因以结合到其后代的基因型中。对于任何正整数N和$ n $，我们精确地解决了在基因型频率参数空间中控制高度非线性耦合微分方程的N个性别匹配$（n + 1）$个等位基因的解析解。作为类比，从两性别到N性别的基因匹配方程的泛化类似于从两体碰撞到N体碰撞的Boltzmann方程，其连续分布函数为离散变量而不是连续变量。我们发现它们在全球范围内的稳定解决方案是一个连续的流形，并且没有混乱。我们的解决方案意味着，在我们的假设下，自然法则不会提供障碍，也不会为支持N性别基因交配稳定系统提供混乱。