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Bifurcations of a two-dimensional discrete-time predator–prey model

Abdul Qadeer Khan

Published 2019 in Advances in Differential Equations

ABSTRACT

We study the local dynamics and bifurcations of a two-dimensional discrete-time predator–prey model in the closed first quadrant 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}. It is proved that the model has two boundary equilibria: O(0,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(0,0)$\end{document}, A(α1−1α1,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A (\frac{\alpha _{1}-1}{\alpha _{1}},0 )$\end{document} and a unique positive equilibrium B(1α2,α1α2−α1−α2α2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B (\frac{1}{\alpha _{2}},\frac{ \alpha _{1}\alpha _{2}-\alpha _{1}-\alpha _{2}}{\alpha _{2}} )$\end{document} under some restriction to the parameter. We study the local dynamics along their topological types by imposing the method of linearization. It is proved that a fold bifurcation occurs about the boundary equilibria: O(0,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(0,0)$\end{document}, A(α1−1α1,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A (\frac{\alpha _{1}-1}{\alpha _{1}},0 )$\end{document} and a period-doubling bifurcation in a small neighborhood of the unique positive equilibrium B(1α2,α1α2−α1−α2α2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B (\frac{1}{\alpha _{2}},\frac{\alpha _{1} \alpha _{2}-\alpha _{1}-\alpha _{2}}{\alpha _{2}} )$\end{document}. It is also proved that the model undergoes a Neimark–Sacker bifurcation in a small neighborhood of the unique positive equilibrium B(1α2,α1α2−α1−α2α2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B (\frac{1}{ \alpha _{2}},\frac{\alpha _{1}\alpha _{2}-\alpha _{1}-\alpha _{2}}{\alpha _{2}} )$\end{document} and meanwhile a stable invariant closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the periodic or quasi-periodic oscillations between predator and prey populations. Numerical simulations are presented to verify not only the theoretical results but also to exhibit the complex dynamical behavior such as the period-2, -4, -11, -13, -15 and -22 orbits. Further, we compute the maximum Lyapunov exponents and the fractal dimension numerically to justify the chaotic behaviors of the discrete-time model. Finally, the feedback control method is applied to stabilize chaos existing in the discrete-time model.

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