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Pulsating waves and entire solutions for a spatially periodic nonlocal dispersal system with a quiescent stage

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Abstract

This paper deals with the pulsating waves and entire solutions for a spatially periodic nonlocal dispersal model with a quiescent stage. By the method of super- and subsolutions together with the comparison principle, we establish the existence of pulsating waves with any speed larger than the spreading speed. The construction of the super- and subsolutions depends on the principal eigenvalue theory for a periodic eigenvalue problem with partially degenerate nonlocal dispersal. The joint results of this paper and our recent work (Wang J B, Li W T, Sun J W. Global dynamics and spreading speeds for a partially degenerate system with non-local dispersal in periodic habitats. Proc Roy Soc Edinburgh Sect A, 2018, 148: 849–880) show that the spreading speed coincides with the minimal wave speed of pulsating waves for the considered system. Moreover, combining the rightward and leftward pulsating waves with different speeds and a spatially periodic solution, we prove the existence and qualitative properties of entire solutions other than pulsating waves, which provide some new spreading ways in a heterogeneous habitat.

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Acknowledgements

This work was supported by the Scientific Research Start-up Fee for High-level Talents (Grant No. 162301182740) and National Natural Science Foundation of China (Grant Nos. 11731005, 11671180 and 11901543). The authors are grateful to the referees for their valuable comments and suggestions which led to an improvement of the original manuscript.

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Authors and Affiliations

  1. School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, China

    Jiabing Wang

  2. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

    Wantong Li

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  1. Jiabing Wang
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  2. Wantong Li
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Correspondence to Wantong Li.

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Wang, J., Li, W. Pulsating waves and entire solutions for a spatially periodic nonlocal dispersal system with a quiescent stage. Sci. China Math. 62, 2505–2526 (2019). https://doi.org/10.1007/s11425-019-1588-1

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  • DOI: https://doi.org/10.1007/s11425-019-1588-1

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