  A nonlinear cross-diffusion--fluid system with chemical terms describing the
dynamics of predator-prey living in a Newtonian fluid is proposed in this
paper. The existence of a weak solution for the proposed macro-scale system is
proved based on the Schauder fixed-point theory, a priori estimates, and
compactness arguments. The proposed system is derived from the underlying
description delivered by a kinetic-fluid theory model by a multiscale approach.
Finally, we discuss the computational results for the proposed macro-scale
system in two-dimensional space.
