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Pattern Dynamics of Cross Diffusion Predator–Prey System with Strong Allee Effect and Hunting Cooperation

Author

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  • Meng Zhu

    (College of Mathematics and Physics, Wenzhou University, Wenzhou 325000, China)

  • Jing Li

    (School of Data Science and Artificial Intelligence, Wenzhou University of Technology, Wenzhou 325000, China)

  • Xinze Lian

    (School of Data Science and Artificial Intelligence, Wenzhou University of Technology, Wenzhou 325000, China)

Abstract

In this paper, we consider a Leslie–Gower cross diffusion predator–prey model with a strong Allee effect and hunting cooperation. We mainly investigate the effects of self diffusion and cross diffusion on the stability of the homogeneous state point and processes of pattern formation. Using eigenvalue theory and Routh–Hurwitz criterion, we analyze the local stability of positive equilibrium solutions. We give the conditions of Turing instability caused by self diffusion and cross diffusion in detail. In order to discuss the influence of self diffusion and cross diffusion, we choose self diffusion coefficient and cross diffusion coefficient as the main control parameters. Through a series of numerical simulations, rich Turing structures in the parameter space were obtained, including hole pattern, strip pattern and dot pattern. Furthermore, We illustrate the spatial pattern through numerical simulation. The results show that the dynamics of the model exhibits that the self diffusion and cross diffusion control not only form the growth of dots, stripes, and holes, but also self replicating spiral pattern growth. These results indicate that self diffusion and cross diffusion have important effects on the formation of spatial patterns.

Suggested Citation

  • Meng Zhu & Jing Li & Xinze Lian, 2022. "Pattern Dynamics of Cross Diffusion Predator–Prey System with Strong Allee Effect and Hunting Cooperation," Mathematics, MDPI, vol. 10(17), pages 1-20, September.
  • Handle: RePEc:gam:jmathe:v:10:y:2022:i:17:p:3171-:d:905708
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    References listed on IDEAS

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    1. Ma, Tingting & Meng, Xinzhu & Hayat, Tasawar & Hobiny, Aatef, 2021. "Stability analysis and optimal harvesting control of a cross-diffusion prey-predator system," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
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    4. Nawaf N. Hamadneh & Muhammad Tahir & Waqar A. Khan, 2021. "Using Artificial Neural Network with Prey Predator Algorithm for Prediction of the COVID-19: The Case of Brazil and Mexico," Mathematics, MDPI, vol. 9(2), pages 1-14, January.
    5. Yan, Xiao & Maimaiti, Yimamu & Yang, Wenbin, 2022. "Stationary pattern and bifurcation of a Leslie–Gower predator–prey model with prey-taxis," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 201(C), pages 163-192.
    6. Liu, Xia & Zhang, Tonghua & Meng, Xinzhu & Zhang, Tongqian, 2018. "Turing–Hopf bifurcations in a predator–prey model with herd behavior, quadratic mortality and prey-taxis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 496(C), pages 446-460.
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    Cited by:

    1. Ying Yu & Yahui Chen & You Zhou, 2023. "Cross-Diffusion-Induced Turing Instability in a Two-Prey One-Predator System," Mathematics, MDPI, vol. 11(11), pages 1-12, May.

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