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

• Published in: Fractal and fractional Vol. 6; no. 1; p. 31
• Main Authors: Tassaddiq, Asifa, Shabbir, Muhammad Sajjad, Din, Qamar, Naaz, Humera
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.01.2022
• Subjects: Bifurcation theory; boundedness; Differential equations; Discretization; Dynamical systems; Equilibrium; fractal dimensions; Fractal geometry; Fractal models; hybrid control; Liapunov exponents; Neimark-Sacker bifurcation; period-doubling bifurcation; Population; Predator-prey simulation; Predators; prey-predator model; Stability analysis
• ISSN: 2504-3110; 2504-3110
• DOI: 10.3390/fractalfract6010031

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.