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