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Traveling waves of predator–prey system with a sedentary predator

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Abstract

The existence of traveling waves to general predator–prey systems is an interesting and ongoing problem, and predator–prey systems that the predator spreads faster than the prey are the major focus of the previous research. In this paper, we are concerned with the existence of traveling waves to predator–prey system with the density death function of the predator when the predator is stationary. First of all, by the unstable manifold theorem, we show that pursuit and evasion waves do not exist for any wave speeds c, as well as invasion waves do not exist for any wave speeds \(c\leqslant 0\). Then, with the aid of the shooting argument, Wazewski theorem, LaSalle’s invariance principle and sophisticated analyses, we establish the existence of invasion waves for wave speed \(c>0\), which partially answers the open problem proposed in Wu (Comput. Math. Appl. 76(5):1139-1160, 2018). Finally, we apply the existence results to Yodzis predator–prey system with Holling type-III functional response and then find a new phenomenon different from previous results.

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The numerical simulations in this paper are carried out using MATLAB.

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Acknowledgements

The authors are grateful to the anonymous referees for their useful suggestions which improve the contents of this paper.

Funding

Yang Wang’s research is supported in part by the National Natural Science Foundation of China (No.11901366) and Shanxi Scholarship Council of China (2021-001). Rong Yuan’s research is supported in part by the National Natural Science Foundation of China (No.12171039 and 12271044). Zhaohai Ma’s research is supported in part by the National Natural Science Foundation of China (No.12001502).

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Authors and Affiliations

  1. Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Hongliang Li & Rong Yuan

  2. School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi, 030006, People’s Republic of China

    Yang Wang

  3. School of Science, China University of Geosciences, Beijing, 100083, People’s Republic of China

    Zhaohai Ma

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  1. Hongliang Li
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Hongliang Li wrote the main manuscript text. All authors reviewed the manuscript.

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Correspondence to Hongliang Li.

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Li, H., Wang, Y., Yuan, R. et al. Traveling waves of predator–prey system with a sedentary predator. Z. Angew. Math. Phys. 74, 198 (2023). https://doi.org/10.1007/s00033-023-02092-9

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  • DOI: https://doi.org/10.1007/s00033-023-02092-9

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