TY - JOUR
T1 - The formation of spreading front
T2 - the singular limit of three-component reaction–diffusion models
AU - Izuhara, Hirofumi
AU - Monobe, Harunori
AU - Wu, Chang Hong
N1 - Funding Information:
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.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2021/4
Y1 - 2021/4
N2 - 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.
AB - 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.
KW - Free boundary problems
KW - Reaction–diffusion systems
KW - Singular limit
KW - Spreading front
KW - Traveling waves
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U2 - 10.1007/s00285-021-01591-5
DO - 10.1007/s00285-021-01591-5
M3 - Article
C2 - 33725162
AN - SCOPUS:85102711342
SN - 0303-6812
VL - 82
JO - Journal of Mathematical Biology
JF - Journal of Mathematical Biology
IS - 5
M1 - 38
ER -