Against the backdrop of global aridification, the spatial self-organization of vegetation in semi-arid regions is critical to ecosystem stability. This study employs mathematical modeling to unravel the synergistic effects of soil-water diffusion feedback (beta) and vegetation internal competition (sigma) on pattern formation. Extending the Klausmeier framework, we integrate a cross-diffusion term Delta(omega-beta n) for root-driven soil-water feedback and a density-dependent competition term 1+ sigma n(2). Key findings include: (1) Increasing.. drives sequential transitions from homogeneous states to gap, stripe, and spot patterns, while rising beta induces reverse transitions, revealing their antagonistic roles in stability; (2) Enhanced precipitation promotes the expansion of discrete spots into continuous stripes, eventually forming gaps under water surplus. Root-mediated water uptake balances short-term facilitation and long-term competition, explaining adaptive responses to resource constraints. Spot patterns act as desertification warnings, whereas stripes offer restoration strategies via high water-use efficiency. By merging cross-diffusion and competition mechanisms, this work proposes a theoretical framework for arid land management, linking mechanistic insights to practical solutions against land degradation.