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Organized periodic structures and coexistence of triple attractors in a predator–prey model with fear and refuge

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  • Garai, Shilpa
  • Pati, N.C.
  • Pal, Nikhil
  • Layek, G.C.

Abstract

We report the existence of periodic structures viz., Arnold tongues, and shrimp-shaped structures in the transitional and chaotic regimes in bi-parameter spaces of a predator–prey model. The model is constructed taking into consideration of two important effects: the prey refuge and fear of predation risk. The fixed points, their existence, and stability behaviors are analyzed. The occurrences of transcritical and Neimark–Sacker bifurcations are studied in both analytical and numerical approaches. The complex dynamical behaviors of the predator–prey model are explored in the bi-parameter space with the help of the largest Lyapunov exponent and isoperiodic diagrams. A completely new kind of U-shaped periodic structure has emerged by merging two Arnold tongues of the same period in the parameter plane. The transition to chaos also occurs through period-bubbling process. The coexistence of triple heterogeneous attractors indicates the unpredictability of the present deterministic model. The basin sets of triple coexisting attractors are drawn, and we observe the presence of Wada basin boundaries. Our studies reveal that the oscillations of the predator–prey populations in certain control parameter regions are highly dependent upon the initial densities of the populations.

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  • 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).
    • Handle: RePEc:eee:chsofr:v:165:y:2022:i:p2:s0960077922010128
    DOI: 10.1016/j.chaos.2022.112833
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    1. 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.
    2. Hossain, Mainul & Pal, Nikhil & Samanta, Sudip, 2020. "Impact of fear on an eco-epidemiological model," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    3. 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).
    4. Sönke Eggers & Michael Griesser & Jan Ekman, 2005. "Predator-induced plasticity in nest visitation rates in the Siberian jay (Perisoreus infaustus)," Behavioral Ecology, International Society for Behavioral Ecology, vol. 16(1), pages 309-315, January.
    5. Zhang, Huisen & Cai, Yongli & Fu, Shengmao & Wang, Weiming, 2019. "Impact of the fear effect in a prey-predator model incorporating a prey refuge," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 328-337.
    6. Shon S. Schooler & Buck Salau & Mic H. Julien & Anthony R. Ives, 2011. "Alternative stable states explain unpredictable biological control of Salvinia molesta in Kakadu," Nature, Nature, vol. 470(7332), pages 86-89, February.
    7. 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).
    8. 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).
    9. Ghosh, Joydev & Sahoo, Banshidhar & Poria, Swarup, 2017. "Prey-predator dynamics with prey refuge providing additional food to predator," Chaos, Solitons & Fractals, Elsevier, vol. 96(C), pages 110-119.
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    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.
    3. Haoming Shi & Fei Xu & Jinfu Cheng & Victor Shi, 2023. "Exploring the Evolution of the Food Chain under Environmental Pollution with Mathematical Modeling and Numerical Simulation," Sustainability, MDPI, vol. 15(13), pages 1-17, June.
    4. Qingyi Cui & Changjin Xu & Wei Ou & Yicheng Pang & Zixin Liu & Peiluan Li & Lingyun Yao, 2023. "Bifurcation Behavior and Hybrid Controller Design of a 2D Lotka–Volterra Commensal Symbiosis System Accompanying Delay," Mathematics, MDPI, vol. 11(23), pages 1-23, November.
    5. Abbasi, Muhammad Aqib & Samreen, Maria, 2024. "Analyzing multi-parameter bifurcation on a prey–predator model with the Allee effect and fear effect," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
    6. Maranhão, Dariel M. & Medrano-T, Rene O., 2024. "Periodicity in the asymmetrical quartic map," Chaos, Solitons & Fractals, Elsevier, vol. 186(C).

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