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Sign of the pulsating wave speed for the bistable competition–diffusion system in a periodic habitat

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Abstract

This paper is concerned with the speeds of pulsating waves for two-species competition–diffusion systems with bistable structures in periodically varying media. The existence and qualitative properties of pulsating waves have been established recently. In this work, assuming that the two species share the same diffusion rates, we study the sign of wave speeds by comparing the reactions and competitions. We first give a criterion when the speed is zero, and then provide some sufficient conditions ensuring the speed has a strict sign. We also show that the spatial heterogeneity has a significant consequence on the sign and gives rise to some new phenomena. More precisely, the pulsating waves in two opposite directions may have different speeds. Particularly, from the ecological point of view, our result indicates that the invasion of a new species may succeed in one direction but fail in the opposite one. Finally, we show the presence of multiple stationary waves which is in contrast with the uniqueness of non-stationary waves.

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Acknowledgements

The authors would like to thank the anonymous referees for careful reading and valuable comments. Weiwei Ding was partly supported by the National Natural Science Foundation of China (12001206) and the Basic and Applied Basic Research Foundation of Guangdong Province (2019A1515110506). Xing Liang was partially supported by the National Natural Science Foundation of China (11971454).

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Weiwei Ding

  2. School of Mathematical Sciences and Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Hefei, 230026, Anhui, China

    Xing Liang

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  1. Weiwei Ding
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Correspondence to Xing Liang.

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Communicated by giga.

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Ding, W., Liang, X. Sign of the pulsating wave speed for the bistable competition–diffusion system in a periodic habitat. Math. Ann. 385, 1–36 (2023). https://doi.org/10.1007/s00208-022-02372-1

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  • DOI: https://doi.org/10.1007/s00208-022-02372-1