Title: On the Simpson index for the Moran process with random selection 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.