Influence of the mode of reproduction on dispersal evolution during species invasion

Vincent Calvez,Frédéric Kuczma,Léonard Dekens,Joachim Crevat,Florian Lavigne,Gaël Raoul,Benoit Fabrèges,Magali Ribot,Eva Lochërbach,Clair Poignard,Nicolas Vauchelet,Céline Grandmont

doi:10.1051/proc/202067008

Abstract

We consider a reaction-diffusion-reproduction equation, modeling a population which is spatially heterogeneous. The dispersion of each individuals is influenced by its phenotype. In the literature, the asymptotic propagation speed of an asexual population has already been rigorously determined. In this paper we focus on the difference between the asexual reproduction case, and the sexual reproduction case, involving a non-local term modeling the reproduction. This comparison leads to a different invasion speed according to the reproduction. After a formal analysis of both cases, leading to a heuristic of the asymptotic behaviour of the invasion fronts, we give some numerical evidence that the acceleration rate of the spatial spreading of a sexual population is slower than the acceleration rate of an asexual one. The main difficulty to get sharper results on a transient comes from the non-local sexual reproduction term.

Highlights

Biological populations are often evolving in a heterogeneous environment, to which individuals can be more or less fitted, according to their phenotypes
Asexual case we present two different numerical schemes to approximate the diffusion with asexual reproduction
As mentioned in the Introduction, some heuristics predict that the sexual reproduction slows down the wave expansion in comparison to the asexual reproduction

Summary

Introduction

Biological populations are often evolving in a heterogeneous environment, to which individuals can be more or less fitted, according to their phenotypes. Similar techniques were developed recently in a different context: the asymptotic description of equilibria in quantitative genetics models involving a sexual mode of reproduction in the regime of small variance. This methodology was recently completed for Fisher’s infinitesimal model [14]. We present formal calculations in order to compare the rates of expansion for the two modes of reproduction under study To validate these heuristics, we present numerical results, which help us to catch a transient regime and its main features

Our model

Formal analysis

Schemes and numerical results

Auto-similar variables

Sexual case

Conclusion