Permanence and existence of periodic solutions for a generalized system with feedback control

Sufficient conditions are obtained for the permanence and the existence of positive periodic solutions of the delay differential system with feedback controlequation(0.1)dxdt=x(t)[F(t,x(t-τ1(t)),…,x(t-τn(t)))-c(t)u(t-δ(t))],dudt=-η(t)u(t)+a(t)x(t-σ(t)).The method involves the application of estimation for uniform upper and lower bounds of solutions. When these results are applied to some special population models with multiple delays, some new results are obtained and some known results are generalized. Especially, our conclusions generalize and complement the results in Chen et al. [F.D. Chen, J.H. Yang, L.J. Chen, X.D. Xie, On a mutualism model with feedback controls, Appl. Math. Comput. 214 (2009) 581–587] and Huo and Li [H.F. Huo, W.T. Li, Positive periodic solutions of a class of delay differential system with feedback control, Appl. Math. Comput. 148 (2004) 35–46].