Agent-Based Validation of the MELV Cooperation Threshold: Version 2.1 (2026)

Agent-based validation of the i = 1 cooperation threshold predicted by the Modified Energetic Lotka-Volterra (MELV) framework. Python/Mesa 3.5 implementation; 81 conditions × 5 replicates = 405 runs at 2000 steps per run. Version 2.1 introduces three biologically motivated architectural improvements over the initial design: (1) a sigmoid Allee effect for cooperator efficiency, requiring a minimum neighbourhood quorum (τ = 0.5) before cooperation becomes self-sustaining — grounded in Allee (1931) and Courchamp et al. (1999); (2) frequency-dependent squared suppression for competitors, (1 − β)², creating positive feedback that stabilises the competitive attractor — grounded in Gore et al. (2009) and Nowak & May (1992); (3) a 20×20 spatial grid reducing stochastic drift artefacts and enabling spatial self-organisation of cooperative clusters. Key results:- Bimodal i-factor distribution confirmed (Hartigan dip p ≈ 0)- Threshold universality across evolutionary rates ε (ANOVA F = 1.91, p = 0.15)- Strong negative correlation mean i vs cooperation level (r = −0.866, p ≈ 0)- φ×β product as near-perfect crossing predictor (sensitivity 1.0, specificity 0.997)- ESS mutant invasion test: 34/34 tests recovered to original basin (100%, both basins), mean invasion displacement Δi = 0.570 — demonstrates two genuine dynamical attractor basins- THRESH residual confirmed as genuine boundary ecology at 2000 steps (9% of runs, late-time drift ≈ 0) Includes 7 figures, full trajectory and summary CSV data, perturbation test results, and a biological correspondences document mapping results to eusociality thresholds, mutualism stability conditions, and cross-taxa cooperation threshold universality. Architecture peer-reviewed by Gemini and Grok (April 2026). Biological fidelity over parameter convenience maintained throughout. Grounded in Blueprint for Harmony (Cooperation Press, 2026; ISBN 978-969-8992-10-1).