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Existence of an extinction wave in the Fisher equation with a shifting habitat

Authors: Haijun Hu and Xingfu Zou
Journal: Proc. Amer. Math. Soc. 145 (2017), 4763-4771
MSC (2010): Primary 34C05, 34D20; Secondary 92D40, 92D25
Published electronically: July 28, 2017
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Abstract: This paper deals with the existence of traveling wave solutions of the Fisher equation with a shifting habitat representing a transition to a devastating environment. By constructing a pair of appropriate upper/lower solutions and using the method of monotone iteration, we prove that for any given speed of the shifting habitat edge, this reaction-diffusion equation admits a monotone traveling wave solution with the speed agreeing to the habitat shifting speed, which accounts for an extinction wave. This predicts not only how fast but also in what manner a biological species will die out in such a shifting habitat.

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Additional Information

Haijun Hu
Affiliation: School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan, People’s Republic of China, 410114

Xingfu Zou
Affiliation: Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada, N6A 5B7

Received by editor(s): October 3, 2016
Published electronically: July 28, 2017
Additional Notes: The first author was partially supported by NNSF of China (Nos. 11326116, 11401051) and Hunan Provincial Natural Science Foundation (Nos. 2015JJ3013, 2016JJ1001). The second author was partially supported by NSERC of Canada (No. RGPIN-2016-04665)
Communicated by: Yingfei Yi
Article copyright: © Copyright 2017 American Mathematical Society

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