Permanence of a Semi-Ratio-Dependent Predator-Prey System with Nonmonotonic Functional Response and Time Delay

Abstract

Sufficient conditions for permanence of a semi-ratio-dependent predator-prey system with nonmonotonic functional response and time delay ${\dot{x}}_1\left(t\right)=x_1\left(t\right)\left[r_1\left(t\right)-a_{11}\left(t\right)x_1\left(t-\tau \left(t\right)\right)-a_{12}\left(t\right)x_2\left(t\right)/\left(m^2+x_1^2\left(t\right)\right)\right],$  ${\dot{x}}_2\left(t\right)=x_2\left(t\right)\left[r_2\left(t\right)-a_{21}\left(t\right)x_2(t)/x_1\left(t\right)\right],$ are obtained, where $x_1\left(t\right)$ and $x_2\left(t\right)$ stand for the density of the prey and the predator, respectively, and $m\ne 0$ is a constant. $\tau \left(t\right)\ge 0$ stands for the time delays due to negative feedback of the prey population.