Bifurcation and complex dynamics of a discrete-time predator-prey system with simplified Monod-Haldane functional response

In this paper, we investigate the dynamics of a discrete-time predator-prey system with simplified Monod-Haldane functional response. The existence and local stability of positive fixed point of the discrete dynamical system is analyzed algebraically. It is shown that the system undergoes a flip bifurcation and a Neimark-Sacker bifurcation in the interior of R+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{2}_{+}$\end{document} by using bifurcation theory. Numerical simulation results not only show the consistence with the theoretical analysis but also display new and interesting dynamical behaviors, including phase portraits, period-11 orbits, attracting invariant circle, cascade of period-doubling bifurcation from period-11 leading to chaos, quasi-periodic orbits, and the sudden disappearance of the chaotic dynamics and attracting chaotic set. The Lyapunov exponents are numerically computed to characterize the complexity of the dynamical behaviors.