Abstract
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.
Highlights
Moran or Wright-Fisher processes are probably the most well known model to study the evolution of a population under various effects
We will consider in this paper this Moran model with immigration and selection as general random process
We consider a population, whose size is constant over time equal to J, composed of S + 1 species
Summary
Moran or Wright-Fisher processes are probably the most well known model to study the evolution of a population under various effects. We will consider in this paper this Moran model with immigration and selection as general random process.
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