Fig. 1

Hysteresis in deterministic vs stochastic descriptions. a Hysteresis loop of the deterministic self-regulatory system (positive roots of Eq. (5)). For values of the control parameter \(b\) below a given threshold, there is a unique stable steady state of low protein \(x\) toward which the system evolves independently of the initial conditions. For input signals above a second threshold, the system evolves toward a unique stable steady state of high \(x\). For signal values within both thresholds, the system is bistable, and evolves toward one stable state or another depending on the initial conditions. In the bistability region, enclosed by two saddle-node bifurcations, three different steady states coexist for a given \(b\) (stable and unstable branches are depicted using solid and dotted lines, respectively). b Transient hysteresis in the stochastic self-regulatory system: slow transients lead to multiple mean states leading to a transitory hysteretic behavior. Red and blue lines are transient solutions obtained from two different initial conditions in the form of Gaussian distributions \({\mathcal{N}}(\mu ,\sigma )\) with mean \(\mu\) and standard deviation \(\sigma\). When the system achieves the stationary state (black solid line corresponds to the stationary solution of the PIDE model), there is a unique mean x-value for given \(b\) (hysteresis disappears). As time increases, the solution gets closer to the stationary distribution. Simulations have been carried out in SELANSI¹⁸