Pattern formation in a vasculogenesis model

This paper investigates 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 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.