Understanding the invasion processes of biological species is a fundamental issue in ecology. Several mathematical models have been proposed to estimate the spreading speed of species. In recent decades, it was reported that some mathematical models of population dynamics have an explicit form of the evolution equations for the spreading front, which are represented by free boundary problems such as the Stefan-like problem (e.g., Mimura et al., Jpn J Appl Math 2:151–186, 1985; Du and Lin, SIAM J Math Anal 42:377–405, 2010). To understand the formation of the spreading front, in this paper, we will consider the singular limit of three-component reaction–diffusion models and give some interpretations for spreading front from the viewpoint of modeling. As an application, we revisit the issue of the spread of the grey squirrel in the UK and estimate the spreading speed of the grey squirrel based on our result. Also, we discuss the relation between some free boundary problems related to population dynamics and mathematical models describing Controlling Invasive Alien Species. Lastly, we numerically consider the traveling wave solutions, which give information on the spreading behavior of invasive species.

了解生物物种的入侵过程是生态学中的一个基本问题。已经提出了几种数学模型来估计物种的传播速度。近几十年来，据报道，一些种群动态的数学模型具有明确形式的传播前沿演化方程，它们由自由边界问题表示，例如 Stefan 类问题（例如，Mimura 等人，Jpn J Appl Math 2:151–186, 1985; Du and Lin, SIAM J Math Anal 42:377–405, 2010)。为了理解扩展前沿的形成，本文将考虑三组分反应-扩散模型的奇异极限，并从建模的角度对扩展前沿进行一些解释。作为应用程序，我们重新审视了英国灰松鼠的传播问题，并根据我们的结果估计了灰松鼠的传播速度。此外，我们还讨论了一些与种群动态相关的自由边界问题与描述控制外来入侵物种的数学模型之间的关系。最后，我们在数值上考虑了行波解，它提供了有关入侵物种传播行为的信息。