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

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.