A two species amensalism model with non-monotonic functional response

Abstract

A two species amensalism model with non-monotonic functional response takes the form

$$\begin{array}{rcl}

\di\frac{dx_1}{dt}&=&x_1\Big(a_1-b_1x_1-\di\frac{c_1x_2 }{d_1+x_2^2}\Big),\\[4mm]

\di\frac{dx_2}{dt}&=&x_2(a_2-b_2x_2),

\end{array}

$$

is proposed and studied, where $a_i, b_i, i=1,2$ $c_1$ and $d_1$ are all positive constants. If $a_1b_2^2d_1+a_1a_2^2-a_2b_2c_1>0$, then the system admits a unique globally stable positive equilibrium, which means that two species could coexistent in a stable state, and if $a_1b_2^2d_1+a_1a_2^2-a_2b_2c_1<0$, then the first species will be driven to extinction, and second species will be convergence to $x_2^*=\frac{r_2}{a_{22}}$}.