Early warning signals for regime shifts in gene transcription regulatory systems: Relaxation time within the perspective of stochastic resonance

Regime shifts frequently occur in bistable dynamical systems, and predicting such transitions is of great importance across various fields. Based on the theory of critical slowing down, increases in autocorrelation and variance have been widely used as early warning signals. However, the reliability of variance may be limited near critical points due to the presence of high-frequency environmental noise or reduced sensitivity of the system to external perturbations. In this study, we propose the relaxation time as a robust early warning signal and systematically evaluate its performance under three types of stochastic noise. We also compared it with the recent dynamic eigenvalue that is rooted in bifurcation theory of dynamical systems to estimate the dominant eigenvalue of the system. Our results show that relaxation time consistently increases prior to critical transition and is more robust than the dynamic eigenvalue. Furthermore, we apply stochastic resonance to explain the observed increase in relaxation time. In this mechanism, system responses to low-frequency perturbations are amplified under appropriate noise conditions, which in turn prolongs its relaxation time. These findings suggest that relaxation time is a simple and reliable indicator for predicting regime shifts.