The method of inside dynamics provides a theory that can track the dynamics of neutral gene fractions in spreading populations. However, the role of mutations has so far been absent in the study of the gene flow of neutral fractions via inside dynamics. Using integrodifference equations, we develop a neutral genetic mutation model by extending a previously established scalar inside dynamics model. To classify the mutation dynamics, we define a mutation class as the set of neutral fractions that can mutate into one another. We show that the spread of neutral genetic fractions is dependent on the leading edge of population as well as the structure of the mutation matrix. Specifically, we show that the neutral fractions that contribute to the spread of the population must belong to the same mutation class as the neutral fraction found in the leading edge of the population. We prove that the asymptotic proportion of individuals at the leading edge of the population spread is given by the dominant right eigenvector of the associated mutation matrix, independent of growth and dispersal parameters. In addition, we provide numerical simulations to demonstrate our mathematical results, to extend their generality and to develop new conjectures about our model.

中文翻译：

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带突变的积分微分方程的内部动力学

内部动力学方法提供了一种理论，可以跟踪传播种群中中性基因部分的动态。然而，迄今为止，在通过内部动力学研究中性部分的基因流中，突变的作用尚不存在。使用积分差方程，我们通过在动力学模型内部扩展先前建立的标量来开发中性基因突变模型。为了对突变动态进行分类，我们将突变类定义为可以相互突变的一组中性分数。我们表明中性遗传分数的传播取决于种群的前沿以及突变矩阵的结构。具体来说，我们表明，有助于种​​群扩散的中性部分必须与在种群前沿发现的中性部分属于相同的突变类别。我们证明在种群传播前沿的个体的渐近比例由相关突变矩阵的显性右特征向量给出，与生长和扩散参数无关。此外，我们提供数值模拟来证明我们的数学结果，扩展它们的普遍性并开发关于我们模型的新猜想。