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The adaptation of biological species to their environment depends on theirtraits. When various biological processes occur (survival, reproduction,migration, etc.), the trait distribution may change with respect to time andspace. In the context of invasions, when considering the evolution of aheritable trait that encodes the dispersive ability of individuals, the traitdistribution develops a particular spatial structure that leads to theacceleration of the front propagation. That phenomenon is known as spatialsorting. Many biological examples can be cited like the bush cricket inBritain, the cane toad invasion in Australia or the common myna one in SouthAfrica. Adopting this framework, recent mathematical studies have led tohighlight the influence of the reproductive mode on the front propagation.Asexual populations have been shown to spread with an asymptotic rate of t 3/2in a minimal reactiondiffusion model, whereas the analogous rate for sexualpopulations is of t 5/4 (where t denotes the time). However, the precisedescription of the behaviour of the front propagation in the sexual case isstill an open question. The aim of this paper is to give precise approximationsfor large times of its position, as well as some features of the local traitdistribution at the front. To do so, we solve explicitly the asymptotic problemderived formally. Numerical simulations are shown to confirm thesecalculations.

Front propagation of a sexual population with evolution of dispersion: a formal analysis