Moran or Wright–Fisher processes are probably the most well known models to study the evolution of a population under environmental various effects. Our object of study will be the Simpson index which measures the level of diversity of the population, one of the key parameters for ecologists who study for example, forest dynamics. Following ecological motivations, we will consider, here, the case, where there are various species with fitness and immigration parameters being random processes (and thus time evolving). The Simpson index is difficult to evaluate when the population is large, except in the neutral (no selection) case, because it has no closed formula. Our approach relies on the large population limit in the “weak” selection case, and thus to give a procedure which enables us to approximate, with controlled rate, the expectation of the Simpson index at fixed time. We will also study the long time behavior (invariant measure and convergence speed towards equilibrium) of the Wright–Fisher process in a simplified setting, allowing us to get a full picture for the approximation of the expectation of the Simpson index.

中文翻译：

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关于随机选择和移民的 Wright-Fisher 过程的 Simpson 指数

Moran 或 Wright-Fisher 过程可能是研究人口在各种环境影响下进化的最著名模型。我们的研究对象将是衡量人口多样性水平的辛普森指数，这是研究森林动态等生态学家的关键参数之一。遵循生态动机，我们将在这里考虑各种物种的适应度和迁移参数是随机过程（因此是时间演变）的情况。当总体很大时，辛普森指数很难评估，除非在中性（无选择）情况下，因为它没有封闭的公式。我们的方法依赖于“弱”选择情况下的大人口限制，因此提供了一个程序，使我们能够以受控的速率近似，Simpson 指数在固定时间的期望值。我们还将在简化设置中研究 Wright-Fisher 过程的长时间行为（不变测度和收敛速度），使我们能够全面了解 Simpson 指数的期望值的近似值。