SIAM Journal on Applied Mathematics, Volume 81, Issue 4, Page 1441-1460, January 2021.
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 highlighting 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 reaction-diffusion model, whereas the analogous rate for sexual populations is of $t^{5/4}$ (where $t$ denotes the time). However, the precise description of the behavior 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.

中文翻译：

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有性人口的前向传播与离散的演变：正式分析

SIAM 应用数学杂志，第 81 卷，第 4 期，第 1441-1460 页，2021 年 1 月。
生物物种对环境的适应取决于它们的特性。当发生各种生物过程（生存、繁殖、迁徙等）时，性状分布可能会随时间和空间发生变化。在入侵的背景下，当考虑编码个体分散能力的遗传性状的进化时，性状分布会形成特定的空间结构，导致前向传播加速。这种现象被称为空间排序。可以引用许多生物学例子，例如英国的丛林蟋蟀、澳大利亚的甘蔗蟾蜍入侵或南非的常见八哥。采用这个框架，最近的数学研究突出了再生模式对前沿传播的影响。在最小反应扩散模型中，无性群体以 $t^{3/2}$ 的渐近速率传播，而有性群体的类似速率为 $t^{5/4}$（其中 $ t$ 表示时间）。但是，对性案例中前向传播行为的准确描述仍然是一个悬而未决的问题。本文的目的是为其位置的大时间提供精确的近似值，以及前端局部特征分布的一些特征。为此，我们明确地解决形式上推导出的渐近问题。显示了数值模拟以确认这些计算。对性案例中前传播行为的准确描述仍然是一个悬而未决的问题。本文的目的是为其位置的大时间提供精确的近似值，以及前端局部特征分布的一些特征。为此，我们明确地解决形式上推导出的渐近问题。显示了数值模拟以确认这些计算。对性案例中前传播行为的准确描述仍然是一个悬而未决的问题。本文的目的是为其位置的大时间提供精确的近似值，以及前端局部特征分布的一些特征。为此，我们明确地解决形式上推导出的渐近问题。显示了数值模拟以确认这些计算。