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Dynamics in the biparametric spaces of a three-species food chain model with vigilance

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  • Hossain, Mainul
  • Kumbhakar, Ruma
  • Pal, Nikhil

Abstract

In nature, vigilance is a common anti-predator strategy prey individuals employ to protect themselves from a possible attack. It gives them sufficient time to take cover or flee. Yet this seemingly profitable strategy, which lowers the number of successful predatory attacks, has a significant impact on the prey's growth rate. Researchers attribute this to reduced foraging time. In this article, we study a three-species food chain model in which both the basal prey and middle predator show active vigilance against their respective predators. Apart from the positivity, boundedness, and local stability analysis of the equilibrium points of the model, we find the conditions for global stability and the occurrence of Hopf bifurcation around the coexisting equilibrium point. Bifurcation diagrams and their corresponding Lyapunov exponent diagrams (largest and second-largest) illustrate diverse dynamical scenarios ranging from stable coexistence to periodic to chaotic oscillations. The isospike diagrams and their associated largest Lyapunov exponent diagrams in the biparametric space of basal prey's and middle predator's vigilance reveal a broader picture of the effects of different levels of vigilance on the dynamics of the system. A great deal of attention has been paid to unwrapping the complex structural beauties like shrimp-shaped periodic structures inside the chaotic sea and multistability between several sets of attractors in the overlapping zones of periodic windows. We observe that multiple coexisting attractors possess fractal basin boundaries. We also explore the dynamics of the system in two other biparametric spaces. The density variation maps at the end show the effect of vigilance on the equilibrium density of all three species. Our results suggest that vigilance promotes regularity and coexistence, but too high levels of vigilance can cause species extinction. For better visualization of the system dynamics, we provide a few high-resolution animations of the phase portraits and the basins of attraction in the supplementary material.

Suggested Citation

  • Hossain, Mainul & Kumbhakar, Ruma & Pal, Nikhil, 2022. "Dynamics in the biparametric spaces of a three-species food chain model with vigilance," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
  • Handle: RePEc:eee:chsofr:v:162:y:2022:i:c:s0960077922006488
    DOI: 10.1016/j.chaos.2022.112438
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    References listed on IDEAS

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    1. da Silva, Rodrigo A. & Rech, Paulo C., 2015. "Spiral periodic structures in a parameter plane of an ecological model," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 9-13.
    2. Pati, N.C. & Layek, G.C. & Pal, Nikhil, 2020. "Bifurcations and organized structures in a predator-prey model with hunting cooperation," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    3. Hossain, Mainul & Pati, N.C. & Pal, Saheb & Rana, Sourav & Pal, Nikhil & Layek, G.C., 2021. "Bifurcations and multistability in a food chain model with nanoparticles," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 808-825.
    4. Scott Creel & Paul Schuette & David Christianson, 2014. "Effects of predation risk on group size, vigilance, and foraging behavior in an African ungulate community," Behavioral Ecology, International Society for Behavioral Ecology, vol. 25(4), pages 773-784.
    5. Tametang Meli, Maxim Idriss & Yemélé, David & Leutcho, Gervais Dolvis, 2021. "Dynamical analysis of series hybrid electric vehicle powertrain with torsional vibration: Antimonotonicity and coexisting attractors," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
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    1. Garai, Shilpa & Pati, N.C. & Pal, Nikhil & Layek, G.C., 2022. "Organized periodic structures and coexistence of triple attractors in a predator–prey model with fear and refuge," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    2. Kumbhakar, Ruma & Hossain, Mainul & Karmakar, Sarbari & Pal, Nikhil, 2024. "An investigation of the parameter space in a tri-trophic food chain model with refuge," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 217(C), pages 37-59.

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