Almost Periodic Solution of a Modified Leslie-Gower Predator-Prey Model with Beddington-DeAngelis Functional Response and Feedback Controls

Abstract

We consider a modified Leslie-Gower predator-prey model with the Beddington-DeAngelis functional response and feedback controls as follows: $\dot{x}\left(t\right)=x\left(t\right)\left(a_1\left(t\right)-b\left(t\right)x\left(t\right)-c\left(t\right)y\left(t\right)/\left(\alpha \left(t\right)+\beta \left(t\right)x\left(t\right)+\gamma \left(t\right)y\left(t\right)\right)-e_1\left(t\right)u\left(t\right)\right)$, $\dot{u}\left(t\right)=-d_1\left(t\right)u\left(t\right)+p_1\left(t\right)x\left(t-\tau \right)$, $\dot{y}\left(t\right)=y\left(t\right)\left(a_2\left(t\right)-r\left(t\right)y\left(t\right)/\left(x\left(t\right)+k\left(t\right)\right)-e_2\left(t\right)\nu \left(t\right)\right)$, and $\dot{\nu }(t)=-d_2(t)\nu (t)+p_2(t)y(t-\tau )$. Sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained.