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Spreading–Vanishing Scenarios in a Time-Periodic Parasitic–Mutualistic Model of Mistletoes and Birds in Heterogeneous Environment with Free Boundary

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Abstract

In this paper, we investigate the asymptotic dynamics of a time-periodic parasitic–mutualistic model of mistletoes and birds in heterogeneous environment with the especial concerns over the spreading–vanishing scenarios, in which the Stefan class free boundary is introduced as the spreading frontier. By defining the ecological reproduction number and generalizing it as the spatial-temporal risk index, a considerably universal spreading–vanishing dichotomy and the sharp criteria are first established in birds world in the absence of mistletoes, and some estimates of the asymptotic spreading speed of the free boundary provided that spreading occurs are also obtained. Furthermore, the comprehensive considerations containing the spreading frontiers, asymptotic profiles and estimates of the asymptotic spreading speed are exhibited in mistletoes-birds world by the monotone iteration technique with the proper upper and lower solutions. The results suggest that even for the spreading case, the mistletoes population will eventually persist in long term provided that its own risk index is larger than 1, otherwise it may be eradicated.

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Acknowledgements

The authors would like to express their sincere thanks to the editor and the anonymous reviewer for their helpful comments and suggestions, which have promoted the original manuscript positively. This research was partially supported by National Natural Science Foundation of China (11701243) and the HongLiu first-class disciplines Development Program of Lanzhou University of Technology.

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  1. Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, 730050, Gansu, People’s Republic of China

    Jie Wang, Jian Wang & Lin Zhao

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  1. Jie Wang
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Correspondence to Jie Wang.

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Wang, J., Wang, J. & Zhao, L. Spreading–Vanishing Scenarios in a Time-Periodic Parasitic–Mutualistic Model of Mistletoes and Birds in Heterogeneous Environment with Free Boundary. J Dyn Diff Equat 35, 1409–1434 (2023). https://doi.org/10.1007/s10884-021-09968-2

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  • DOI: https://doi.org/10.1007/s10884-021-09968-2

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