On the Simpson index for the Moran process with random selection and immigration

TL;DR

This paper develops a forward approximation method for the Simpson index in Moran and Wright-Fisher processes with random, evolving fitness and immigration parameters, enabling better understanding of biodiversity dynamics in large populations.

Contribution

It introduces a novel forward approach to approximate the expected Simpson index in large populations with random parameters, extending beyond neutral cases and previous backward methods.

Findings

Provides a controlled approximation procedure for the Simpson index over time.

Analyzes the long-term behavior of the Wright-Fisher process with random parameters.

Offers insights into biodiversity dynamics in ecological models with evolving fitness and immigration.

Abstract

Moran or Wright-Fisher processes are probably the most well known model to study the evolution of a population under various effects. Our object of study will be the Simpson index which measures the level of diversity of the population, one of the key parameter 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). To measure biodiversity, ecologists generally use the Simpson index, who has no closed formula, except in the neutral (no selection) case via a backward approach, and which is difficult to evaluate even numerically when the population size is large. Our approach relies on the large population limit in the "weak" selection case, and thus to give a procedure which enable us to approximate, with controlled rate, the expectation of the Simpson index at fixed time. Our approach will be forward and valid for all time, which is the main difference with the historical approach of Kingman, or Krone-Neuhauser. We will also study the long time behaviour 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.