Dynamic behaviors of a nonlinear amensalism model

Abstract

Abstract A nonlinear amensalism model of the form dN1dt=r1N1(1−(N1P1)α1−u(N2P1)α2),dN2dt=r2N2(1−(N2P2)α3), $$\begin{aligned} &\frac{dN_{1}}{dt}= r_{1}N_{1} \biggl(1- \biggl( \frac{N_{1}}{P_{1}} \biggr)^{\alpha _{1}}-u \biggl(\frac{N_{2}}{P_{1}} \biggr)^{\alpha_{2}} \biggr), \\ &\frac{dN_{2}}{dt}= r_{2}N_{2} \biggl(1- \biggl( \frac{N_{2}}{P_{2}} \biggr)^{\alpha_{3}} \biggr), \end{aligned}$$ where ri,Pi,u,i=1,2,α1,α2,α3 $r_{i}, P_{i}, u, i=1, 2, \alpha_{1}, \alpha_{2}, \alpha_{3}$ 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(P2P1)α2 $1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}} $. If 1−u(P2P1)α2>0 $1-u (\frac {P_{2}}{P_{1}} )^{\alpha_{2}}>0$, then the unique positive equilibrium D(N1∗,N2∗) $D(N_{1}^{*},N_{2}^{*})$ is globally attractive, if 1−u(P2P1)α2<0 $1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}<0$, then the boundary equilibrium C(0,P2) $C(0, P_{2})$ 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).

Keywords

• Amensalism model
• Differential inequality theory
• Global stability