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We present Graph Neural ODEs (GNODEs) for modeling tumor microenvironment dynamics with mathematically guaranteed stability and conservation properties. Unlike bulk ODEs that miss spatial heterogeneity or discrete GNNs that inadequately capture continuous biological processes, GNODEs provide continuous-time evolution with explicit adjacency-aware dynamics while maintaining provable trajectory bounds. Our framework ensures: (1) existence and uniqueness of solutions under dynamic graph topology, (2) Lyapunov stability preventing unphysical states like negative cell counts, and (3) exact conservation of biological invariants through architectural constraints. Benchmarking on synthetic tumor data demonstrates that GNODE accurately captures resistant cell fraction dynamics (0.282 predicted vs 0.242 true) while graph-free alternatives fail completely (0.000), proving that stability-constrained local interactions are essential for modeling emergent resistance.