Permanence and periodic solutions for an impulsive reaction–diffusion food-chain system with ratio-dependent functional response

An impulsive reaction–diffusion periodic food-chain system with ratio-dependent functional response is investigated in the present paper. Sufficient conditions for the ultimate boundedness and permanence of the food-chain system are established based on the comparison theory of differential equation and upper and lower solution method. By constructing appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given in the end of the paper.