Asymptotic behavior of the solution to a competitive system with feedback controls and functional reaction response (Q2735466)

The authors consider the following two species competitive nonautonomous system with functional response and feedback controls NEWLINE\[NEWLINE\begin{aligned} \dot{x}_1(t)&= x_1(t)[b_1(t)-a_{11}(t)x_1(t)-\frac{a_{12}(t)x_2(t)}{1+ \alpha (t)x_1(t)}-c_1(t)u_1(t)],\\ \dot{x}_2(t)&= x_2(t)[b_2(t)-a_{22}(t)x_2(t)-\frac{a_{21}(t)x_1(t)}{1+ \alpha (t)x_1(t)}+c_2(t)u_2(t)],\\ \dot{u}_1(t)&= -e_1(t)u_1(t)+d_1(t)x_1(t),\\ \dot{u}_2(t)&= f(t)-e_2(t)u_2(t)-d_2(t)x_2(t), \end{aligned}NEWLINE\]NEWLINE where \(b_i(t)\), \(a_{ij}(t)\), \(e_i(t)\), \(d_i(t)\), \(f(t)\), \(\alpha(t)\), \(i, j= 1,2\), are continuous functions which are bounded from above and from below by positive constants. Some sufficient conditions for the persistence are obtained. Furthermore, if the system is periodic, then it has a strictly positive periodic solution which is globally asymptotically stable under appropriate conditions.