Hopf-Bifurcation and Pattern Selections in a Three Trophic Level Food Web System

The ecosystem comprises many food webs, and their existence is directly dependent upon the growth rate of primary prey; it balances the whole ecosystem. This paper studies the temporal and spatiotemporal dynamics of three trophic levels of food web system, consisting of two preys and two predators. We first obtained an equilibrium solution set and studied the system’s stability at a biological feasible equilibrium point using a Jacobian method. We show the occurrence of Hopf-bifurcation by considering the growth rate of prey as the bifurcation parameter for the temporal model. In the presence of diffusion, we study random movement in species, establish conditions for the system’s stability, and derive the Turing instability condition. A multiple-scale analysis is used to determine the amplitude equations in the neighborhood of the Turing bifurcation point. After applying amplitude equations, the system has a rich dynamical behavior. The stability analysis of these amplitude equations leads to the development of various Turing patterns. Finally, with numerical simulations, the analytical results are verified. Within this framework, our study through the dynamical behavior of the complex system and bifurcation point based on the prey growth rate can serve as a baseline for numerous researchers working on ecological models from diverse perspectives. As a result, the Hopf-bifurcation and multiple-scale analysis used in the complex food web system is particularly relevant experimentally because the linked consequences may be researched and applied to many mathematical, ecological, and biological models.