Dynamic behaviors of a nonlinear amensalism model

A nonlinear amensalism model of the form $$\begin{aligned} &\frac{dN_{1}}{dt}= r_{1}N_{1} \biggl(1- \biggl( \frac{N_{1}}{P_{1}} \biggr)^{\alpha _{1}}-u \biggl(\frac{N_{2}}{P_{1}} \biggr)^{\alpha_{2}} \biggr), \\ &\frac{dN_{2}}{dt}= r_{2}N_{2} \biggl(1- \biggl( \frac{N_{2}}{P_{2}} \biggr)^{\alpha_{3}} \biggr), \end{aligned}$$ where \(r_{i}, P_{i}, u, i=1, 2, \alpha_{1}, \alpha_{2}, \alpha_{3}\) are all positive constants, is proposed and studied in this paper. The dynamic behaviors of the system are determined by the sign of the term \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}} \). If \(1-u (\frac {P_{2}}{P_{1}} )^{\alpha_{2}}>0\), then the unique positive equilibrium \(D(N_{1}^{*},N_{2}^{*})\) is globally attractive, if \(1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}<0\), then the boundary equilibrium \(C(0, P_{2})\) is globally attractive. Our results supplement and complement the main results of Xiong, Wang, and Zhang (Advances in Applied Mathematics 5(2):255–261, 2016).