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Effects of nonlinear growth, cross-diffusion and protection zone on a diffusive predation model

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Abstract

This paper concerns a diffusive predation model with nonlinear growth, cross-diffusion and protection zone terms. The main purpose is to investigate the effects of nonlinear growth and cross-diffusion on the coexistent solution when protection zone is present. Firstly, a priori estimate and the existence of positive solutions are discussed, including local and global existence. Then, some asymptotic properties of coexistent solutions induced by the mortality rate, nonlinear growth of predator and cross-diffusion are analyzed. It is revealed that there exist critical values related to certain principal eigenvalues such that the nonlinear growth, cross-diffusion and protection zone all have significant effects on the coexistent solutions; as far as the nonlinear growth concerned, we find that it has important influences on the coexistence region of two species undoubtedly. Biologically, this implies that these critical values greatly affect the survival of species.

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Funding

The work is supported in part by the National Natural Science Foundations of China (12171296), the Natural Science Basic Research Program of Shaanxi (2024JC-YBQN-0006) and the Youth Innovation Team of Shaanxi Universities.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, 710062, Shaanxi, China

    Daoxin Qiu & Yunfeng Jia

  2. School of Mathematics and Data Sciences, Changji University, Changji, 831100, Xinjiang, China

    Yunfeng Jia

  3. School of Mathematics, Xi’an University of Finance and Economics, Xi’an, 710100, Shaanxi, China

    Jingjing Wang

Authors
  1. Daoxin Qiu
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  2. Yunfeng Jia
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  3. Jingjing Wang
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Contributions

Daoxin Qiu contributed to conceptualization, writing—original draft, investigation, software, and formal analysis. Yunfeng Jia contributed to conceptualization, methodology, investigation, writing—review, and formal analysis. Jingjing Wang contributed to conceptualization, investigation, and formal analysis. All authors reviewed the manuscript

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Correspondence to Yunfeng Jia.

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Qiu, D., Jia, Y. & Wang, J. Effects of nonlinear growth, cross-diffusion and protection zone on a diffusive predation model. Z. Angew. Math. Phys. 75, 109 (2024). https://doi.org/10.1007/s00033-024-02254-3

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  • DOI: https://doi.org/10.1007/s00033-024-02254-3

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