A model for the spatio-temporal evolution of three biological species in a food chain model consisting of two competitive preys and one predator with intra-specific competition is considered. Besides diffusing, the predator species moves toward higher concentrations of a chemical substance produced by the prey. The prey, in turn, moves away from high concentrations of a substance secreted by the predators. The resulting reaction-diffusion system consists of three parabolic equations along with three elliptic equations describing the diffusion of the chemical substances. The local existence of nonnegative solutions is proved. Then uniform estimates in Lebesgue spaces are provided. These estimates lead to boundedness and global well-posedness for the system. Numerical simulations are presented and discussed. (c) 2022 Elsevier Ltd. All rights reserved.