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The effects of diffusion on the dynamics of a Lotka-Volterra predator-prey model with a protection zone

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Abstract

This paper is concerned with the dynamics of a diffusive predator-prey model with a protection zone under homogeneous Neumann boundary conditions. By assuming that the diffusion rates of two species are one, Du and Shi (JDE 229:63–91, 2006) studied the impact of protection zone on the dynamics and derived significant difference of the model’s behavior from the original no-protection zone model. The main purpose of this paper is to remove this assumption and investigate the impact of different diffusion rates and protection zone on the dynamics of the model. The sufficient conditions for the existence and non-existence of positive steady states in term of the diffusion rates are established. More importantly, the limiting profiles of positive steady states (when they exist) are also derived as the diffusion rates tend to 0 or \(\infty \). Our research here provides important information about how the diffusion rates affect the dynamics of the model.

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Acknowledgements

We sincerely thank an anonymous reviewer for his (her) helpful comments and suggestions. The work was partially supported by NSF of China (11901446,11971369,12171381,12171296), the Postdoctoral Science Foundation of China (2021T140530), and the Young Talent fund of University Association for Science and Technology in Xi’an (095920201325).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, People’s Republic of China

    Shanbing Li

  2. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, People’s Republic of China

    Jianhua Wu

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  1. Shanbing Li
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  2. Jianhua Wu
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Correspondence to Shanbing Li.

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Communicated by A. Neves.

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Li, S., Wu, J. The effects of diffusion on the dynamics of a Lotka-Volterra predator-prey model with a protection zone. Calc. Var. 61, 213 (2022). https://doi.org/10.1007/s00526-022-02338-w

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