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The study of non-constant steady states and pattern formation for an interacting population model in a spatial environment

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  • Gupta, R.P.
  • Tiwari, Shristi
  • Kumar, Arun

Abstract

This manuscript accounts for an investigation of the complex dynamics of a spatial model for interacting populations. We discuss the existence and boundedness of solutions for the proposed spatio-temporal system. The global stability of the co-existing steady state of the proposed system is analyzed with the help of a suitable Lyapunov function. We provide results on the existence and non-existence of positive non-constant solutions of the model. The priori estimate for the positive steady state is obtained for the nonexistence of the non-constant positive steady state by using the maximum principle. The existence of a non-constant positive steady state is studied with the help of Leray–Schauder degree theory. The stability and Hopf bifurcation are briefly revisited for the co-existing steady state in the corresponding temporal model, where a bubble-like structure is observed. The onset of Hopf bifurcation has been analyzed, and different conditions for the formation of the Turing pattern have been established through diffusion-driven instability analysis. Numerical simulations are performed in detail to figure out the effects of saturated harvesting on Turing patterns. The Turing as well as non-Turing patterns in their respective domains are also examined. Finally, the criteria of Turing–Hopf bifurcation is briefly demonstrated with relevant numerical examples and corresponding plots that give a better illustration of this work.

Suggested Citation

  • Gupta, R.P. & Tiwari, Shristi & Kumar, Arun, 2025. "The study of non-constant steady states and pattern formation for an interacting population model in a spatial environment," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 229(C), pages 652-672.
    • Handle: RePEc:eee:matcom:v:229:y:2025:i:c:p:652-672
    DOI: 10.1016/j.matcom.2024.10.022
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    References listed on IDEAS

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    1. Wu, Daiyong & Yang, Youwei & Wu, Peng, 2023. "Impacts of prey-taxis and nonconstant mortality on a spatiotemporal predator–prey system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 283-300.
    2. Chen, Mengxin & Wu, Ranchao, 2023. "Steady states and spatiotemporal evolution of a diffusive predator–prey model," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    3. Kumar, Vikas & Kumari, Nitu, 2021. "Bifurcation study and pattern formation analysis of a tritrophic food chain model with group defense and Ivlev-like nonmonotonic functional response," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).
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