从随机的、基于个体的模型到自适应动力学的正则方程-一步到位
From stochastic, individual-based models to the canonical equation of adaptive dynamics - In one step
摘要：
我们考虑一个模型的达尔文进化的无性种群与一个大的，但不恒定的人口规模的特点是自然出生率，逻辑死亡率建模竞争和突变的概率在每个出生事件。在本文中，我们研究了系统的长期行为的限制，大人口$(K\to \infty)$大小，罕见的突变$(u\to 0)$，和小的突变效应$(σ\to 0)$，证明收敛到正则方程的自适应动力学（CEAD）。与Champagnat和Méléard等人的早期研究相反，我们同时取三个极限，即$u=u_K$和$σ=σ_K$，在$K$的情况下趋于零，条件是确保出生和死亡事件的时间尺度与成功突变事件的时间尺度保持分离。这减慢了微观系统的动力学，并导致严重的技术困难，需要使用完全不同的方法。特别是，我们不能用固定所需发散时间的大数定律来用相应的确定性系统来近似随机系统。为了解决这个问题，我们开发了一个“随机欧拉方案”的基础上耦合参数，允许控制时间尺度上的随机系统的时间演化与$K$发散。
英文摘要：
We consider a model for Darwinian evolution in an asexual population with a large but non-constant populations size characterized by a natural birth rate, a logistic death rate modelling competition and a probability of mutation at each birth event. In the present paper, we study the long-term behavior of the system in the limit of large population $(K\to \infty)$ size, rare mutations $(u\to 0)$, and small mutational effects $(σ\to 0)$, proving convergence to the canonical equation of adaptive dynamics (CEAD). In contrast to earlier works, e.g. by Champagnat and Méléard, we take the three limits simultaneously, i.e. $u=u_K$ and $σ=σ_K$, tend to zero with $K$, subject to conditions that ensure that the time-scale of birth and death events remains separated from that of successful mutational events. This slows down the dynamics of the microscopic system and leads to serious technical difficulties that requires the use of completely different methods. In particular, we cannot use the law of large numbers on the diverging time needed for fixation to approximate the stochastic system with the corresponding deterministic one. To solve this problem we develop a "stochastic Euler scheme" based on coupling arguments that allows to control the time evolution of the stochastic system over time-scales that diverge with $K$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
