# The effect of competition on the neutral intraspecific diversity of invasive species

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## Abstract

This paper deals with the effect of interspecific competition on the dynamics of neutral genetic diversity in a range-expanding population. The spread of an invasive species in an environment already hosting a resident competitor is described by a traveling wave solution with minimal speed, \(u(t,x)=U(x - c^* \, t)\), of a diffusive Lotka–Volterra competition model. The description of the dynamics of neutral genetic fractions in this wave is based on a decomposition of the wave into several components, as proposed by Roques et al. (Proc Natl Acad Sci USA 109(23):8828–8833, 2012). Our analytical results reveal that the wave can be either the pulled type, corresponding to strong erosion of the diversity, or the pushed type, corresponding to maintenance of the initial diversity. The pulled/pushed nature of the wave depends on the linear or nonlinear nature of the speed \(c^*\). Our results show that, for sufficiently strong competition, the speed is nonlinear, and therefore all of the genetic diversity in the invasive population is maintained. Conversely, in the absence of competition, or when competition is mild, the speed is linear, which means that only the furthest forward fraction in the initial invasive population eventually remains in the colonization front. Our numerical results also show that the sufficient conditions of Lewis et al. (J Math Biol 45(3):219–233, 2002) and Huang (J Dyn Differ Equ 22(2):285–297, 2010) for the linearity of the speed \(c^*\) can still be improved, and they show that nonlinear speeds occur across a wide region of the parameter space, providing a counterpoint to recent analytical results suggesting that nonlinear speeds only occur in certain limiting cases.

## Keywords

Lotka–Volterra Competition Genetic diversity Pulled and pushed waves Nonlinear speed## Mathematics Subject Classification

35C07 35Q92 35K45 35K55 35K57## Notes

### Acknowledgments

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement N. 321186—ReaDi—Reaction–Diffusion Equations, Propagation and Modelling, and from the French Agence Nationale pour la Recherche, within the ANR-10-INTB-1705-04-MACBI project. The second author was supported by JSPS KAKENHI Grant Number 22540156 and the ReaDiLab (LIA 197, CNRS).

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