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Effects of chemotaxis and time delay on the spatiotemporal patterns of a two-species reaction–diffusion system

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  • Zuo, Wenjie
  • Song, Binbin
  • Chen, Yuming

Abstract

In this paper, we investigate the effects of chemotaxis and time delay on the spatiotemporal dynamics of a two-species reaction–diffusion system. We show that cheomotaxis and delay can induce the instability of the constant steady state and the existence of a stable spatially non-homogeneous steady state bifurcation, spatially homogeneous/non-homogeneous Hopf bifurcation , and double Hopf bifurcation. Particularly, for the case of no delay, we drive the formula for determining the direction and stability of the degenerate steady state bifurcation. Furthermore, we apply our theoretical results to a delayed predator–prey system with predator-taxis and a cooperative Lotka–Volterra system. For the predator–prey system, predator-taxis and delay jointly lead to spatially non-homogeneous periodic solutions due to spatially non-homogeneous Hopf bifurcation and double-Hopf bifurcation via the interaction between Hopf bifurcations. For the cooperative system, spatially non-homogeneous steady state solutions and non-homogeneous period solutions bifurcate from the constant equilibrium by Turing bifurcation induced by the chemotaxis term and Hopf bifurcation induced by the delay, though the equilibrium of the corresponding ODE is globally asymptotically stable.

Suggested Citation

  • Zuo, Wenjie & Song, Binbin & Chen, Yuming, 2025. "Effects of chemotaxis and time delay on the spatiotemporal patterns of a two-species reaction–diffusion system," Chaos, Solitons & Fractals, Elsevier, vol. 190(C).
    • Handle: RePEc:eee:chsofr:v:190:y:2025:i:c:s0960077924012888
    DOI: 10.1016/j.chaos.2024.115736
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    References listed on IDEAS

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    1. Wang, Caiyun, 2015. "Rich dynamics of a predator–prey model with spatial motion," Applied Mathematics and Computation, Elsevier, vol. 260(C), pages 1-9.
    2. Pati, N.C. & Ghosh, Bapan, 2022. "Delayed carrying capacity induced subcritical and supercritical Hopf bifurcations in a predator–prey system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 195(C), pages 171-196.
    3. Daiyong Wu & Hai Zhang & Jinde Cao & Tasawar Hayat, 2013. "Stability and Bifurcation Analysis of a Nonlinear Discrete Logistic Model with Delay," Discrete Dynamics in Nature and Society, Hindawi, vol. 2013, pages 1-7, December.
    4. Tiwari, Vandana & Tripathi, Jai Prakash & Mishra, Swati & Upadhyay, Ranjit Kumar, 2020. "Modeling the fear effect and stability of non-equilibrium patterns in mutually interfering predator–prey systems," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    5. Chen, Mengxin & Zheng, Qianqian, 2022. "Predator-taxis creates spatial pattern of a predator-prey model," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    6. Banda, Heather & Chapwanya, Michael & Dumani, Phindile, 2022. "Pattern formation in the Holling–Tanner predator–prey model with predator-taxis. A nonstandard finite difference approach," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 196(C), pages 336-353.
    7. Chen, Mengxin & Zheng, Qianqian, 2023. "Steady state bifurcation of a population model with chemotaxis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    8. Chen, Mengxin & Srivastava, Hari Mohan, 2023. "Stability of bifurcating solution of a predator–prey model," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    9. Chang, Lili & Sun, Gui-Quan & Wang, Zhen & Jin, Zhen, 2015. "Rich dynamics in a spatial predator–prey model with delay," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 540-550.
    10. Song, Binbin & Zuo, Wenjie, 2023. "Spatiotemporal dynamics of a three-component chemotaxis model for Alopecia Areata," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
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