In this talk, we present our recent work on equivariant Turing-Turing bifurcations and pattern formation in square spatial domains. For general reaction-diffusion systems with self-diffusion and cross-diffusion terms, we derive formulated results for the normal form and its coefficients on center manifolds near equivariant Turing-Turing singularities. Through analysis of the normal forms, we give approximate expressions for superposition-type steady states and their stability conditions in the original system. As an application, we analyze a plant-water interaction model to explore the formation of self-organized vegetation patterns in semi-arid regions.