Bifurcations, Permanence and Local Behavior of the Plant-Herbivore Model with Logistic Growth of Plant Biomass

This paper investigates the plant-herbivore model’s dynamics. The plant’s biomass without herbivores growth with logistic equation assuming that the herbivore (parasitization) occurs after the host’s density-dependent growth regulation occurs. We give a topological classification of the equilibrium points. We show that the boundary equilibrium undergoes the transcritical, fold, and period-doubling bifurcation, whereas the interior equilibrium undergoes a Neimark-Sacker bifurcation. We use the OGY method to control chaos produced by period-doubling bifurcation. The system exhibits bistability between the stable interior attractors in the interior and the stable attractors in the x-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x-$$\end{document}boundary logistic dynamics (periodic orbits and strange attractors) for particular numerical values of parameters. Sufficient conditions for the permanence of the plant-herbivores system are obtained, ensuring the coexistence of both species.