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Transactions of the American Mathematical Society

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Propagation dynamics of a time periodic and delayed reaction-diffusion model without quasi-monotonicity


Authors: Liang Zhang, Zhi-Cheng Wang and Xiao-Qiang Zhao
Journal: Trans. Amer. Math. Soc. 372 (2019), 1751-1782
MSC (2010): Primary 35B40, 35K57, 34K05; Secondary 46E25, 20C20
DOI: https://doi.org/10.1090/tran/7709
Published electronically: March 25, 2019
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Abstract: In this paper, we consider a time periodic non-monotone and nonlocal delayed reaction-diffusion population model with stage structure. We first prove the existence of the asymptotic speed $ c^*$ of spread by virtue of two auxiliary equations and comparison arguments. By the method of super- and sub-solutions and the fixed point theorem, as applied to the truncated problem on a finite interval, and the limiting arguments, we then establish the existence of time periodic traveling wave solutions of the model system with wave speed $ c>c^*$. We further use the results of the asymptotic speed of spread to obtain the non-existence of traveling wave solutions for wave speed $ c<c^*$. Finally, we prove the existence of the critical periodic traveling wave with wave speed $ c=c^*$. It turns out that the asymptotic speed of spread coincides with the minimal wave speed for positive periodic traveling waves. These results are also applied to the model system with two prototypical birth functions.


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Additional Information

Liang Zhang
Affiliation: School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China
Email: lz@lzu.edu.cn

Zhi-Cheng Wang
Affiliation: School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China
Email: wangzhch@lzu.edu.cn

Xiao-Qiang Zhao
Affiliation: Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7, Canada
Email: zhao@mun.ca

DOI: https://doi.org/10.1090/tran/7709
Keywords: Nonlocal population model, time delay, non-monotone, asymptotic speed of spread, periodic traveling waves, minimal wave speed
Received by editor(s): September 6, 2017
Received by editor(s) in revised form: May 25, 2018
Published electronically: March 25, 2019
Additional Notes: The first author was supported in part by NNSF of China (11701242) and the Fundamental Research Funds for the Central Universities (lzujbky-2017-27).
The second author was supported in part by NNSF of China (11371179, 11731005) and the Fundamental Research Funds for the Central Universities (lzujbky-2017-ot09). The second author is the corresponding author
The third author was supported in part by the NSERC of Canada.
Zhi-Cheng Wang is the corresponding author
Article copyright: © Copyright 2019 American Mathematical Society