Cross-diffusion systems are nonlinear parabolic systems that describe the diffusion interactions of multi-component agents. In population dynamics, they model the evolution and avoidance behaviors of interacting populations. From a modeling perspective, cross-diffusion terms naturally arise in the fast reaction limit of a rescaled system with linear diffusion and fast reaction. In this talk, we rigorously establish the existence of weak solutions for a triangular cross-diffusion system driven by starvation and dietary diversity, obtained as its singular fast reaction limit. Our approach relies on a priori estimates obtained by the analysis of a family of entropy functionals and compactness arguments. Finally, we conclude with improved regularity and weak-strong stability results.