  We study a chemotaxis system that includes two competitive prey and one
predator species in a two-dimensional domain, where the movement of prey (resp.
predators) is driven by chemicals secreted by predators (resp. prey), called
mutually repulsive (resp. mutually attractive) chemotaxis effect. The kinetics
for all species are chosen according to the competitive Lotka-Volterra
equations for prey and to a Holling type functional response for the predator.
Under the biologically relevant scenario that the chemicals diffuse much faster
than the individual diffusion of all species and a suitable re-scaling,
equations for chemical concentrations are parabolic with slow evolution of
coefficient $0<\varepsilon\ll 1$. In the first main result, we show the global
existence of a unique classical solution to the system for each $\varepsilon$.
Secondly, we study rigorously the so-called fast signal diffusion limit,
passing from the system including parabolic equations with the slow evolution
to the one with all elliptic equations for chemical concentrations, i.e. the
limit as $\varepsilon \to 0$. This explains why elliptic equations can be
proposed for chemical concentration instead of parabolic ones with slow
evolution. Thirdly, the $L^\infty$-in-time convergence rates for the fast
signal diffusion limit are estimated, where the effect of the initial layer is
carefully treated. Finally, the differences between the systems with and
without the slow evolution, and between the systems with one or two preys are
discussed due to numerical simulations.
