Discretization, Bifurcation, and Control for a Class of Predator-Prey Interactions

Abstract

The present study focuses on the dynamical aspects of a discrete-time Leslie–Gower predator–prey model accompanied by a Holling type III functional response. Discretization is conducted by applying a piecewise constant argument method of differential equations. Moreover, boundedness, existence, uniqueness, and a local stability analysis of biologically feasible equilibria were investigated. By implementing the center manifold theorem and bifurcation theory, our study reveals that the given system undergoes period-doubling and Neimark–Sacker bifurcation around the interior equilibrium point. By contrast, chaotic attractors ensure chaos. To avoid these unpredictable situations, we establish a feedback-control strategy to control the chaos created under the influence of bifurcation. The fractal dimensions of the proposed model are calculated. The maximum Lyapunov exponents and phase portraits are depicted to further confirm the complexity and chaotic behavior. Finally, numerical simulations are presented to confirm the theoretical and analytical findings.

The Qualitative Analysis of Host–Parasitoid Model with Inclusion of Spatial Refuge Effect

“…We use this technique to reduce chaos emerging because of the Neimark-Sacker bifurcation. As we know that system (2) experiences the Neimark-Sacker bifurcation at the equilibrium point Ω(x , y ), then a controlled system corresponding to this system can be expressed as [23][24][25][26]:…”

Complex Dynamics of a Predator-Prey System With Gompertz Growth and Herd Behavior

“…Improper management of bifurcations can lead to disturbances in population dynamics and the consequent destruction of ecosystems. For a detailed bifurcation analysis, we recommend that readers refer to [31,[33][34][35][36][37][38][39].…”

“…Discrete-time dynamical systems have complicated and diversified dynamical properties [25][26][27][28][29][30][31]. The system to be analyzed in this work is then obtained by using the forward Euler technique on the system (1.3) as follows:…”

A Modified Iterative Approach for Fixed Point Problem in Hadamard Spaces

“…Its usefulness mostly hinges on the availability of solutions to mathematical problems generated from systems engineering [4] and computer science [5]. Because of its novel development as a confluence of analysis [6,7] and geometry [8], the theory of fixed points has also become a powerful and vital instrument for the study of nonlinear problems [9]. As such, a choice between several distinct iteration approaches must be made, taking important aspects into account.…”