Stability and bifurcation analysis of two species amensalism model with Michaelis–Menten type harvesting and a cover for the first species

In this paper, a two species amensalism model with Michaelis–Menten type harvesting and a cover for the first species that takes the form dx(t)dt=a1x(t)−b1x2(t)−c1(1−k)x(t)y(t)−qE(1−k)x(t)m1E+m2(1−k)x(t),dy(t)dt=a2y(t)−b2y2(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} &\frac{dx(t)}{dt}=a_{1}x(t)-b_{1}x^{2}(t)-c_{1}(1-k)x(t)y(t)- \frac{qE(1-k)x(t)}{m_{1}E+m_{2}(1-k)x(t)}, \\ &\frac{dy(t)}{dt}=a_{2}y(t)-b_{2}y^{2}(t) \end{aligned}$$ \end{document} is investigated, where ai\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{i}$\end{document}, bi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{i}$\end{document}, i=1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$i=1,2$\end{document}, and c1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c_{1}$\end{document} are all positive constants, k is a cover provided for the species x, and 0<k<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< k<1$\end{document}. The stability and bifurcation analysis for the system are taken into account. The existence and stability of all possible equilibria of the system are investigated. With the help of Sotomayor’s theorem, we can prove that there exist two saddle-node bifurcations and two transcritical bifurcations under suitable conditions.