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Asymptotic behavior of the principal eigenvalue and the basic reproduction ratio for periodic patch models

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Abstract

This paper is devoted to the study of the asymptotic behavior of the principal eigenvalue and the basic reproduction ratio associated with periodic population models in a patchy environment for small and large dispersal rates. We first deal with the eigenspace corresponding to the zero eigenvalue of the connectivity matrix. Then we investigate the limiting profile of the principal eigenvalue of an associated periodic eigenvalue problem as the dispersal rate goes to zero and infinity, respectively. We further establish the asymptotic behavior of the basic reproduction ratio in the case of small and large dispersal rates. Finally, we apply these results to a periodic Ross-Macdonald patch model.

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Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant No. 11901138) and the Natural Science Foundation of Shandong Province (Grant No. ZR2019QA006). The second author was supported by the National Sciences and Engineering Research Council of Canada. The authors are grateful to the anonymous referees for their careful reading and helpful comments, which led to an improvement of our original manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai, 264209, China

    Lei Zhang

  2. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada

    Lei Zhang & Xiao-Qiang Zhao

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  1. Lei Zhang
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  2. Xiao-Qiang Zhao
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Correspondence to Lei Zhang.

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Zhang, L., Zhao, XQ. Asymptotic behavior of the principal eigenvalue and the basic reproduction ratio for periodic patch models. Sci. China Math. 65, 1363–1382 (2022). https://doi.org/10.1007/s11425-021-1894-2

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  • DOI: https://doi.org/10.1007/s11425-021-1894-2

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