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Effects of the random walk and the maturation period in a diffusive predator–prey system with two discrete delays

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  • Bilazeroğlu, Ş.
  • Göktepe, S.
  • Merdan, H.

Abstract

This study aims to present a complete Hopf bifurcation analysis of a model describing the relationship between prey and predator. A ratio-dependent reaction–diffusion system with two discrete time delays operating under Neumann boundary conditions governs the model that represents this competition. The bifurcation parameter for the analysis is a delay parameter that reflects the amount of time needed for the predator to be able to hunt. Bilazeroğlu and Merdan’s algorithm (Bilazeroğlu et al., 2021), which is developed by using the center manifold theorem and normal form theory, is used to establish the existence of Hopf bifurcations and also the stability of the bifurcating periodic solutions. The same procedure is used to illustrate some specific bifurcation properties, such as direction, stability, and period. Furthermore, by examining a model with constant coefficients, we also analyze how diffusion and the amount of time needed for prey to mature impact the model’s dynamics. To support the obtained analytical results, we also run some numerical simulations. The results indicate that the dynamic of the mathematical model is significantly influenced by diffusion, the amount of time needed for the predator to gain the capacity to hunt, and the amount of time required for prey to reach maturity that the predator can hunt.

Suggested Citation

  • Bilazeroğlu, Ş. & Göktepe, S. & Merdan, H., 2023. "Effects of the random walk and the maturation period in a diffusive predator–prey system with two discrete delays," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    • Handle: RePEc:eee:chsofr:v:176:y:2023:i:c:s0960077923010020
    DOI: 10.1016/j.chaos.2023.114101
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    References listed on IDEAS

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    1. Wang, Fatao & Yang, Ruizhi, 2023. "Spatial pattern formation driven by the cross-diffusion in a predator–prey model with Holling type functional response," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    2. Karaoglu, Esra & Merdan, Huseyin, 2014. "Hopf bifurcations of a ratio-dependent predator–prey model involving two discrete maturation time delays," Chaos, Solitons & Fractals, Elsevier, vol. 68(C), pages 159-168.
    3. D. Brockmann & L. Hufnagel & T. Geisel, 2006. "The scaling laws of human travel," Nature, Nature, vol. 439(7075), pages 462-465, January.
    4. Bilazeroğlu, Ş. & Merdan, H., 2021. "Hopf bifurcations in a class of reaction-diffusion equations including two discrete time delays: An algorithm for determining Hopf bifurcation, and its applications," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    5. Hu, Guang-Ping & Li, Wan-Tong & Yan, Xiang-Ping, 2009. "Hopf bifurcations in a predator–prey system with multiple delays," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1273-1285.
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