In this paper, a discrete ratio-dependent food-chain system with delay is investigated. By using Gaines and Mawhin’s continuation theorem of coincidence degree theory and the method of Lyapunov function, a set of sufficient conditions for the existence of positive periodic solutions and global asymptotic stability of the model are established. National Natural Science Foundation of China 61673008 Project of High-level Innovative Talents of Guizhou Province [2016]5651 Department of Education of Guizhou Province [2017]039 Project of Key Laboratory of Guizhou Province with Financial and Physical Features [2017]004 Changsha University of Science and Technology 2018MMAEZD21 University Science and Technology Top Talents Project of Guizhou Province KY[2018]047 Guizhou University of Finance and Economics 2018XZD01 Education Department of Hunan Province 17A181
