Global asymptotic stability of periodic solution for a cooperative system with time delays (Q1886961)

The authors consider the delayed cooperative system \[ \begin{aligned} x_1^{\prime }(t) &=r_1(t)x_1(t)\left[ 1-\frac{x_1(t-\tau _{11}(t))}{ a_1(t)+b_1(t)x_2(t-\tau _{12}(t))}-c_1(t)x_1(t-\tau _{11}(t))\right], \\ x_2^{\prime }(t) &=r_2(t)x_2(t)\left[ 1-\frac{x_2(t-\tau _{22}(t))}{ a_2(t)+b_2(t)x_1(t-\tau _{21}(t))}-c_2(t)x_2(t-\tau _{22}(t))\right],\end{aligned} \] \[ x_i(\theta ) =\phi _i(\theta ),\;\theta \in [-\tau ,0],\;\phi _i(0)>0,\;\tau =\max \{\tau _{ij}(t),\;t\in [0,\omega ]\},\;i,j=1,2, \] where \(\phi _i\) is a given nonnegative and bounded continuous function on \( [-\tau ,0].\) By using continuation technique, they prove that the system has at least a positive periodic solution. Furthermore, sufficient conditions are derived for the global stability of the positive periodic solution by constructing a suitable Lyapunov function.