Global attractivity in a competition system with feedback controls.
The authors consider a two species Lotka-Volterra competition system with feedback controls. They derive sufficient conditions for the existence of a globally attracting positive equilibrium in the cases where the indirect controls act instantaneously or with a fixed time delay \(\tau\). These cases are modeled respectively by the systems \[ \begin{aligned} {dx_1(t)\over dt} &=x_1(t)\{b_1-a_{11}x_1(t)-a_{12}x_2(t)-\alpha_1 u_1(t)\},\\ {dx_2(t)\over dt} &=x_2(t)\{b_2-a_{21}x_1(t)-a_{22}x_2(t)-\alpha_2 u_2(t)\},\\ {du_1(t)\over dt} &=-\eta_1u_1(t)+a_1x_1(t),\\ {du_2(t)\over dt} &=-\eta_2u_2(t)+a_2x_2(t),\\ \text{and} {dx_1(t)\over dt} &=x_1(t)\{b_1-a_{11}x_1(t)-a_{12}x_2(t)-\alpha_1 u_1(t-\tau)\},\\ {dx_2(t)\over dt} &=x_2(t)\{b_2-a_{21}x_1(t)-a_{22}x_2(t)-\alpha_2 u_2(t-\tau)\},\\ {du_1(t)\over dt} &=-\eta_1u_1(t)+a_1x_1(t-\tau),\\\;{du_2(t)\over dt} &=-\eta_2u_2(t)+a_2x_2(t-\tau). \end{aligned} \]
scientific article; zbMATH DE number 2017218