Asymptotic profile in selection-mutation equations: Gauss versus Cauchy distributions

TL;DR

This paper investigates the long-term behavior of a population model with small mutation rates, revealing that the population distribution can be Gaussian-like or Cauchy-like depending on the mutation rate scaling.

Contribution

It demonstrates how the asymptotic distribution of phenotypic traits transitions between Gaussian and Cauchy types based on mutation rate scaling in a selection-mutation model.

Findings

Population densities are Gaussian-like for small scaling exponents.

Population densities are Cauchy-like for large scaling exponents.

The interplay between mutation rate and time scale determines the distribution shape.

Abstract

In this paper, we study the asymptotic (large time) behavior of a selection-mutation-competition model for a population structured with respect to a phenotypic trait, when the rate of mutation is very small. We assume that the reproduction is asexual, and that the mutations can be described by a linear integral operator. We are interested in the interplay between the time variable $t$ and the rate $\varepsilon$ of mutations. We show that depending on $\alpha > 0$, the limit $\varepsilon \to 0$ with $t = \varepsilon^{-\alpha}$ can lead to population number densities which are either Gaussian-like (when $\alpha$ is small) or Cauchy-like (when $\alpha$ is large).