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The formation of spreading front: the singular limit of three-component reaction–diffusion models

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Abstract

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.

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Acknowledgements

We appreciate the valuable comments and suggestions of the reviewers and the editor, which help us to improve the manuscript. H. Izuhara was partially supported by JSPS KAKENHI Grant No. 17K14237. H. Monobe was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) (No. 15K17595). C.-H. Wu was partly supported from the Young Scholar Fellowship Program by Ministry of Science and Technology (MOST) in Taiwan, under Grant MOST 109-2636-M-009-008.

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Authors and Affiliations

  1. Faculty of Engineering, University of Miyazaki, 1-1 Gakuen Kibanadai Nishi, Miyazaki, 889-2192, Japan

    Hirofumi Izuhara

  2. Research Institute for Interdisciplinary Science, Okayama University, 3-1-1 Tsushima-naka, Kita-ku, Okayama, 700-8530, Japan

    Harunori Monobe

  3. Department of Applied Mathematics, National Yang Ming Chiao Tung University, Hsinchu, 30010, Taiwan

    Chang-Hong Wu

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  1. Hirofumi Izuhara
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  2. Harunori Monobe
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  3. Chang-Hong Wu
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Correspondence to Chang-Hong Wu.

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Izuhara, H., Monobe, H. & Wu, CH. The formation of spreading front: the singular limit of three-component reaction–diffusion models. J. Math. Biol. 82, 38 (2021). https://doi.org/10.1007/s00285-021-01591-5

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  • DOI: https://doi.org/10.1007/s00285-021-01591-5

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