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A free boundary problem for a ratio-dependent predator–prey system

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Abstract

In this paper, we study a free boundary problem for a ratio-dependent predator–prey system in one space dimension, in which the free boundary is only caused by prey, representing the spreading fronts of prey. We discuss the long time behaviors of solution as \(t\rightarrow \infty \) and establish a spreading–vanishing dichotomy; namely, either the two species successfully spread to infinity as \(t\rightarrow \infty \) and survive in the new environment, or they cannot spread to the whole space and the prey will vanish eventually. Then, the criteria for spreading and vanishing are obtained. Furthermore, when spreading occurs, some estimates of the asymptotic speed of h(t) as \(t\rightarrow \infty \) are provided. Finally, some realistic and meaningful phenomena are discovered.

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Acknowledgements

The author’s work was supported by the National Natural Science Foundation of China 12071316. The author would like to thank Professor Mingxin Wang who gave me invaluable lectures on free boundary. The author is also grateful for the anonymous reviewers for their helpful comments and suggestions.

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Authors and Affiliations

  1. School of Science, Civil Aviation Flight University of China, Guanghan, 618307, People’s Republic of China

    Lingyu Liu

  2. Division of Mathematics, Sichuan University Jinjiang College, Pengshan, 620860, People’s Republic of China

    Chunyan Yang

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  1. Lingyu Liu
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  2. Chunyan Yang
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Correspondence to Lingyu Liu.

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Liu, L., Yang, C. A free boundary problem for a ratio-dependent predator–prey system. Z. Angew. Math. Phys. 74, 69 (2023). https://doi.org/10.1007/s00033-023-01957-3

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