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

R.P. Gupta, Shristi Tiwari and Arun Kumar

Mathematics and Computers in Simulation (MATCOM), 2025, vol. 229, issue C, 652-672

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.

Keywords: Spatio-temporal model; Global stability; Non-constant solutions; Turing instability (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:229:y:2025:i:c:p:652-672

DOI: 10.1016/j.matcom.2024.10.022

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