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Reaction-diffusion predator-prey-parasite system and spatiotemporal complexity

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  • Chakraborty, Bhaskar
  • Ghorai, Santu
  • Bairagi, Nandadulal

Abstract

This paper deals with the spatial pattern formation in a diffusive predator-prey-parasite (PPP) model, where predator feeds on infected prey following type II response function and infection spreads among prey species through horizontal transmission. The study is accomplished with respect to an ecological parameter that quantifies the reproductive gain of predator and two epidemiological parameters which measure the force of infection and virulence of the disease. We show analytically that the interior equilibrium loses its stability through Hopf bifurcation in the absence of diffusion if the reproductive gain of predator crosses some threshold value. In case of diffusive system, it is shown that the interior equilibrium, which is otherwise stable, may lose its stability due to diffusion. Criteria for the occurrence of various types of instability, like Turing, Hopf-Turing and pure Hopf, associated with the PPP model are presented with illustrations. Our simulation results reveal that this diffusion-driven instability creates various spatio-temporal patterns, like spot, stripe, mixture of spots & stripes and spiral patterns, depending upon the values of ecological and diffusion parameters. Turing instability and the corresponding patterns are also observed with the variation of two epidemiological parameters. Interestingly, the epidemiological parameters that measure the infection rate and virulence of the disease show opposite patterns with their increasing values.

Suggested Citation

  • Chakraborty, Bhaskar & Ghorai, Santu & Bairagi, Nandadulal, 2020. "Reaction-diffusion predator-prey-parasite system and spatiotemporal complexity," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    • Handle: RePEc:eee:apmaco:v:386:y:2020:i:c:s0096300320304768
    DOI: 10.1016/j.amc.2020.125518
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    References listed on IDEAS

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    1. Peng, Yahong & Ling, Heyang, 2018. "Pattern formation in a ratio-dependent predator-prey model with cross-diffusion," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 307-318.
    2. Sahoo, Banshidhar, 2015. "Role of additional food in eco-epidemiological system with disease in the prey," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 61-79.
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    2. Chakraborty, Bhaskar & Marick, Sounov & Bairagi, Nandadulal, 2024. "Diffusion-driven instabilities in a tri-trophic food web model: From Turing to non-Turing patterns and waves," Chaos, Solitons & Fractals, Elsevier, vol. 189(P1).
    3. Bhunia, Bidhan & Ghorai, Santu & Kar, Tapan Kumar & Biswas, Samir & Bhutia, Lakpa Thendup & Debnath, Papiya, 2023. "A study of a spatiotemporal delayed predator–prey model with prey harvesting: Constant and periodic diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    4. Ghorai, Santu & Chakraborty, Bhaskar & Bairagi, Nandadulal, 2021. "Preferential selection of zooplankton and emergence of spatiotemporal patterns in plankton population," Chaos, Solitons & Fractals, Elsevier, vol. 153(P1).
    5. Majee, Suvankar & Jana, Soovoojeet & R., Anand & Ramprabhakar, J., 2026. "Complex spatiotemporal dynamics in an eco-epidemic model: Turing instability, non-stationary patterns, and chaos control," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 240(C), pages 650-667.
    6. Sarangi, B.P. & Raw, S.N., 2023. "Dynamics of a spatially explicit eco-epidemic model with double Allee effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 241-263.
    7. Jingen Yang & Zhong Zhao & Yingying Kong & Jing Xu, 2025. "Spatiotemporal Dynamics of a Predator–Prey Model with Harvest and Disease in Prey," Mathematics, MDPI, vol. 13(15), pages 1-22, July.

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