Stability and bifurcation analysis of two species amensalism model with Michaelis–Menten type harvesting and a cover for the first species
 作者： Yu Liu,  Liang Zhao,  Xiaoyan Huang,  Hang Deng 作者单位： 1College of Mathematics and Computer Science, Fuzhou University2College of Information and Statistics, Guangxi University of Finance and Economics 刊名： Advances in Difference Equations, 2018, Vol.2018 (1), pp.1-19 来源数据库： Springer Journal DOI： 10.1186/s13662-018-1752-2 关键词： Amensalism model;  Nonlinear harvesting;  Cover;  Bifurcation analysis; 英文摘要： Abstract(#br)In this paper, a two species amensalism model with Michaelis–Menten type harvesting and a cover for the first species that takes the form d x ( t ) d t = a 1 x ( t ) − b 1 x 2 ( t ) − c 1 ( 1 − k ) x ( t ) y ( t ) − q E ( 1 − k ) x ( t ) m 1 E + m 2 ( 1 − k ) x ( t ) , d y ( t ) d t = a 2 y ( t ) − b 2 y 2 ( t ) \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} is investigated, where a i $a_{i}$ ,... b i $b_{i}$ , i = 1 , 2 $i=1,2$ , and c 1 $c_{1}$ are all positive constants, k is a cover provided for the species x , and 0 < k < 1 $0< k<1$ . 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.