Resolving Stability Conflicts via Geometric Stochastic Dynamics

Traulsen et al. (2006) reveal a paradox: infinite-population stability (Nash equilibria) contradicts finite-population stochastic effects. This research bridges this gap by applying geometric time-reversal theory (O'Byrne & Cates, 2024) to evolutionary dynamics. Specifically, we model population states as points on a manifold where "vorticity" quantifies stability conflicts. For instance, in predator-prey systems (Tomilova et al., 2025), reversibility conditions could identify when stochastic effects dominate deterministic equilibria. Unlike Traulsen et al.'s frequency-dependent Moran process, this approach uses coordinate-free geometric invariants to classify stability regimes. The impact includes a unified theory for stochastic stability across scales, with applications in econophysics (Ryu & Lee, 2016) or neural decision-making (Schilling et al., 2023).

References:

1. Stochasticity and evolutionary stability.. A. Traulsen, J. M. Pacheco, L. Imhof (2006). Physical review. E, Statistical, nonlinear, and soft matter physics.
2. Geometric theory of (extended) time-reversal symmetries in stochastic processes: I. Finite dimension. J. O'Byrne, M. E. Cates (2024). Journal of Statistical Mechanics: Theory and Experiment.
3. In search of balance: the Lotka–Volterra model in biosystem research. Nadezhda M. Tomilova, Olga A. Kishkinova, Natalya A. Verezubova, Natalia E. Sakovich (2025). Veterinariya, Zootekhniya i Biotekhnologiya.
4. Econophysics, Statistical Mechanics for Financial Applications, and Financial Mathematics. Doojin Ryu, Kiseop Lee (2016).
5. Predictive coding and stochastic resonance as fundamental principles of auditory phantom perception. A. Schilling, W. Sedley, Richard C. Gerum, C. Metzner, Konstantin Tziridis, Andreas K. Maier, H. Schulze, F. Zeng, K. Friston, P. Krauss (2023). Brain : a journal of neurology.