Analysis of steady-state solutions to a vasculogenesis model in dimension two

We investigate the steady state solutions of a vasculogenesis model governed by coupled partial differential equations in a bounded two dimensional domain. Explicit steady state solutions are analytically constructed, and their stability is rigorously analyzed under prescribed initial and boundary conditions. By employing the energy method, we prove that these solutions exhibit local asymptotic stability when specific parametric criteria are satisfied. The analysis establishes a direct connection between the stability thresholds and the system's diffusion coefficient, offering quantitative insights into the mechanisms governing pattern formation. These results provide foundational theoretical advances for understanding self organization in chemotaxis-driven biological systems, particularly vasculogenesis.