Distributions of the First Passage Time in a Bistable Biological System

It has been shown that a biological system with interlinked positive and negative feedback loops has the potential to generate rich dynamical behaviors such as monostability, bistability, oscillation, and excitability. Here, we demonstrate that tuning different feedback strengths can lead to the system generating bistability of two different types: two stable steady states, or both a stable steady state and a stable limit cycle. In particular, computing the distribution of the first passage time (FPT) between the two attractors for each type using the stochastic simulation algorithm based on the master equation, we find that the FPT obeys a single-peak distribution if the stochastic trajectories cross the separatrix of the saddle point, and a multi-peak distribution if they across the separatrix of the unstable limit cycle. Such a remarkable difference indicates that the FPT is an effective index for identifying the type of bistability.