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Stability of bifurcating solution of a predator–prey model

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  • Chen, Mengxin
  • Srivastava, Hari Mohan

Abstract

To explore the role of the prey-taxis in an ecological model, we investigate a predator–prey model with prey-taxis in this paper. Firstly, the local stability of the positive equilibrium and the occurrence conditions of the steady state bifurcation are given. Thereafter, we investigate the existence and stability of the bifurcating solution around the threshold. Precisely, by treating the prey-taxis constant ξ as the bifurcation parameter, we confirm the model possesses the steady state bifurcation at ξ=ξkS for k∈N0/{0}. Also, we set ξkS(ɛ)=ξkS+ɛξ1+ɛ2ξ2+⋅⋅⋅ for small ɛ>0. We show that ξ1=0 and ξ2 determines the stability of the bifurcating solution. Finally, the stable bifurcating solution is observed by using numerical experiments. The findings of this paper are: (i) the repulsive prey-taxis will facilitate the occurrence of the steady state bifurcation. (ii) the bifurcating solution is stable if ξ2<0 and it is unstable if ξ2>0.

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  • Chen, Mengxin & Srivastava, Hari Mohan, 2023. "Stability of bifurcating solution of a predator–prey model," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    • Handle: RePEc:eee:chsofr:v:168:y:2023:i:c:s0960077923000541
    DOI: 10.1016/j.chaos.2023.113153
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    References listed on IDEAS

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    1. Chen, Mengxin & Zheng, Qianqian, 2022. "Predator-taxis creates spatial pattern of a predator-prey model," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    2. Yan, Xiao & Maimaiti, Yimamu & Yang, Wenbin, 2022. "Stationary pattern and bifurcation of a Leslie–Gower predator–prey model with prey-taxis," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 201(C), pages 163-192.
    3. 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.
    4. 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).
    5. 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).
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