How long does it take for an initially advantageous mutant to establish itself in a resident population, and what does the population composition look like then? We approach these questions in the framework of the so called Bare Bones evolution model (Klebaner et al. in J Biol Dyn 5(2):147-162, 2011. https://doi.org/10.1080/17513758.2010.506041) that provides a simplified approach to the adaptive population dynamics of binary splitting cells. As the mutant population grows, cell division becomes less probable, and it may in fact turn less likely than that of residents. Our analysis rests on the assumption of the process starting from resident populations, with sizes proportional to a large carrying capacity K. Actually, we assume carrying capacities to be [Formula: see text] and [Formula: see text] for the resident and the mutant populations, respectively, and study the dynamics for [Formula: see text]. We find conditions for the mutant to be successful in establishing itself alongside the resident. The time it takes turns out to be proportional to [Formula: see text]. We introduce the time of establishment through the asymptotic behaviour of the stochastic nonlinear dynamics describing the evolution, and show that it is indeed [Formula: see text], where [Formula: see text] is twice the probability of successful division of the mutant at its appearance. Looking at the composition of the population, at times [Formula: see text], we find that the densities (i.e. sizes relative to carrying capacities) of both populations follow closely the corresponding two dimensional nonlinear deterministic dynamics that starts at a random point. We characterise this random initial condition in terms of the scaling limit of the corresponding dynamics, and the limit of the properly scaled initial binary splitting process of the mutant. The deterministic approximation with random initial condition is in fact valid asymptotically at all times [Formula: see text] with [Formula: see text].

中文翻译：

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关于建立突变体。

一个最初有利的突变体需要多长时间才能在常住人口中建立自己的地位，那么人口结构将是什么样？我们在所谓的裸骨进化模型（Klebaner等人，《生物生物学》（J Biol Dyn）5（2）：147-162，2011. https://doi.org/10.1080/17513758.2010.506041）的框架中解决这些问题。为二进制分裂细胞的自适应种群动态提供了一种简化的方法。随着突变种群的增长，细胞分裂的可能性变得越来越小，实际上它的发生几率也要低于居民。我们的分析基于从居民人口开始的过程假设，其大小与较大的承载能力K成正比。实际上，我们假定居民和居民的承载能力分别为[公式：请参见文字]和[公式：请参见文字]。突变人群 分别研究[公式：参见文字]的动力学。我们找到了使突变体与居民成功建立自己的条件。事实证明，所花费的时间与[公式：参见文字]成正比。我们通过描述演化的随机非线性动力学的渐近行为来介绍建立时间，并证明确实是[公式：参见文本]，其中[公式：参见文本]是突变体成功分裂的概率的两倍。它的外观。有时观察种群的组成[公式：参见文字]，我们发现两个种群的密度（即相对于承载能力的大小）紧密地遵循从随机点开始的相应的二维非线性确定性动力学。我们根据相应动力学的标度极限和突变体的适当标度的初始二元分裂过程的极限来表征这种随机初始条件。实际上，具有随机初始条件的确定性逼近在任何时候都可以渐近有效[公式：参见文本]与[公式：参见文本]。