  We rigorously prove the passage from a Lotka-Volterra reaction-diffusion
system towards a cross-diffusion system at the fast reaction limit. The system
models a competition of two species, where one species has a more diverse diet
than the other. The resulting limit gives a cross-diffusion system of a
starvation driven type. We investigate the linear stability of homogeneous
equilibria of those systems and rule out the possibility of Turing instability.
Numerical simulations are included which are compatible with the theoretical
results.
