Analysis of the Stability and Chaotic Dynamics of an Ecological Model

Abstract

Modelling has become an eminent tool in the study of ecological systems. Ecological modelling can help implement sustainable development, mathematical models, and system analysis that explain how ecological processes can promote the sustainable management of resources. In this paper, we also chose a four-dimensional discrete-time Lotka–Volterra ecological model and analyzed its dynamic behavior. In particular, we derived the parametric conditions for the existence of biologically feasible solutions and the stability of the fixed points. We also provided graphs to study the spectrum behavior of all fixed points. In addition, we have seen that when the intrinsic dynamics of the population exceed a certain threshold, the system bifurcates. This particular range of inherent population dynamics depends on the values of other biological parameters and the initial population. We proved that the instability of the model resulted in Neimark–Sacker and period-doubling bifurcations. To confirm these two types of bifurcation, we used bifurcation theory, and to find the direction of bifurcation, we used graphical results. Mainly, through novel periodic plots, we confirm the coexistence of the population and the possible equilibrium states. We apply Marotto’s theorem to verify the existence of chaos in the system. To control the chaos, we use a hybrid control feedback methodology. Finally, we provide numerical examples to illustrate our theoretical results. The outcomes of the numerical simulations show chaotic long-term behavior across an extensive range of parameters.