Fundamental properties of macroscopic gene-mating dynamic evolutionary systems are investigated. A model is studied to describe a large class of systems within population genetics. We focus on a single locus, any number of alleles in a two-gender dioecious population. Our governing equations are time-dependent continuous differential equations labeled by a set of parameters, where each parameter stands for a population percentage carrying certain common genotypes. The full parameter space consists of all allowed parameters of these genotype frequencies. Our equations are uniquely derived from four fundamental assumptions within any population: (1) a closed system; (2) average-and-random mating process (mean-field behavior); (3) Mendelian inheritance; and (4) exponential growth and exponential death. Even though our equations are nonlinear with time-evolutionary dynamics, we have obtained an exact analytic time-dependent solution and an exactly solvable model. Our findings are summarized from phenomenological and mathematical viewpoints. From the phenomenological viewpoint, any initial parameter of genotype frequencies of a closed system will eventually approach a stable fixed point. Under time evolution, we show (1) the monotonic behavior of genotype frequencies, (2) any genotype or allele that appears in the population will never become extinct, (3) the Hardy–Weinberg law and (4) the global stability without chaos in the parameter space. To demonstrate the experimental evidence for our theory, as an example, we show a mapping from the data of blood type genotype frequencies of world ethnic groups to our stable fixed-point solutions. From the mathematical viewpoint, our highly symmetric governing equations result in continuous global stable equilibrium solutions: these solutions altogether consist of a continuous curved manifold as a subspace of the whole parameter space of genotype frequencies. This fixed-point manifold is a global stable attractor known as the Hardy–Weinberg manifold, attracting any initial point in any Euclidean fiber bounded within the genotype frequency space to the fixed point where this fiber is attached. The stable base manifold and its attached fibers form a fiber bundle, which fills in the whole genotype frequency space completely. We can define the genetic distance of two populations as their geodesic distance on the equilibrium manifold. In addition, the modification of our theory under the process of natural selection and mutation is addressed.

中文翻译：

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基因交配动态进化理论：基本假设，可精确求解的模型和解析解。

研究了宏观基因交配动态进化系统的基本特性。研究了一个模型来描述种群遗传学中​​的一大类系统。我们关注一个单一的基因座，一个两性雌雄异体的人群中的任何等位基因。我们的控制方程是由一组参数标记的随时间变化的连续微分方程，其中每个参数代表携带某些常见基因型的总体百分比。完整的参数空间由这些基因型频率的所有允许参数组成。我们的方程式是从任何总体中的四个基本假设中唯一得出的：（1）封闭系统；（2）平均和随机的交配过程（平均场行为）；（3）孟德尔遗传；（4）指数增长和指数死亡。即使我们的方程是具有时间演化动力学的非线性方程，我们也获得了精确的时变解析解和可解模型。我们的发现是从现象学和数学观点总结的。从现象学的角度来看，封闭系统的基因型频率的任何初始参数最终都将接近稳定的固定点。在时间演化下，我们证明（1）基因型频率的单调性；（2）种群中出现的任何基因型或等位基因都不会灭绝；（3）Hardy-Weinberg定律；（4）全局稳定性而不会造成混乱在参数空间中。为了证明我们的理论的实验证据，例如，我们显示了从世界种族的血型基因型频率数据到我们的稳定定点解的映射。从数学观点来看，我们的高度对称的控制方程导致连续的全局稳定平衡解：这些解总共由一个连续的弯曲流形组成，作为基因型频率整个参数空间的子空间。该定点流形是一个称为Hardy-Weinberg流形的全局稳定吸引子，它将在基因型频率空间内界定的任何欧几里得纤维中的任何起始点吸引到该光纤所连接的固定点。稳定的基础歧管及其连接的纤维形成一个纤维束，该束完全填充了整个基因型频率空间。我们可以将两个种群的遗传距离定义为平衡流形上的测地距离。此外，还讨论了在自然选择和突变过程中对我们理论的修改。