Note on the persistent property of a feedback control system with delays

By developing some new analysis techniques, we show that the following feedback control system of differential equations with delays is permanent. dN(t)dt=r(t)N(t)[1−N2(t−τ1(t))k2(t)−c(t)u(t−τ2(t))],du(t)dt=−a(t)u(t)+b(t)N(t−τ1(t)), where τ1,τ2,a,b,c,r,k∈C(R,(0,+∞))τ1,τ2,a,b,c,r,k∈C(R,(0,+∞)) are ωω-periodic functions, which means that feedback control variable has no influence on the persistent property of the above system. Our result supplements the main result of Fan, Li and Qin [G.H. Fan, Y.K. Li, M.C. Qin, The existence of positive periodic solutions for periodic feedback control systems with delays, Zeitschrift für Angewandt Mathematik und Mechanik 84 (6) (2004) 425–430].