Node-Density Network Model for Multi-Species Evolutionary and Ecological Dynamics: Analytical Foundations and Empirical Validation

Understanding coupled genetic and ecological dynamics across biological scales remains a central unsolved problem. Existing gene regulatory network (GRN) models evolve network architecture but omit population ecology; eco-evolutionary models couple ecology to evolution via trait distributions but treat molecular gene networks as black boxes. No existing framework simultaneously resolves gene regulatory dynamics, derives individual fitness from network state, and feeds that fitness into coupled multi-species population equations within a single ODE system. The Node-Density Network Model (NDNM) closes this gap. We derive two analytical results from NDNM: (1) a closed-form Evolutionary Rescue Threshold giving necessary and sufficient conditions for population survival under sustained environmental stress as a function of gene network parameters; and (2) a Fitness-Mediated Coexistence Condition linking gene expression equilibria to multi-species community stability. We validate NDNM against the Lenski Long-Term Evolution Experiment (LTEE) in E. coli, demonstrating that fitted NDNM parameters reproduce the empirically observed hyperbolic fitness trajectory across 50,000 generations. Together, these results establish NDNM as a mathematically rigorous and empirically grounded framework for eco-evolutionary dynamics.