Time triggered stochastic hybrid systems with nonlinear continuous dynamics

Abstract

Time-triggered stochastic hybrid systems (TTSHS) constitute a class of piecewise-deterministic Markov processes (PDMP), where continuous-time evolution of the state space is interspersed with discrete stochastic events. Whenever a stochastic event occurs, the state space is reset based on a random map. Prior work on this topic has focused on the continuous-time evolution being modeled as a linear time-invariant system. In this contribution, we generalize those results to a system with nonlinear continuous dynamics. Our approach relies on approximating the nonlinear dynamics between two successive events as a linear time-varying system and using this approximation to derive analytical solutions for the state space’s statistical moments. The TTSHS framework is used to model continuous growth in an individual cell’s size and subsequent division into daughters. It is well known that exponential growth in cell size, together with a size-independent division rate, leads to an unbounded variance in cell size. Motivated by recent experimental findings, we consider nonlinear growth in cell size based on a Michaelis-Menten function and show that this leads to size homeostasis in the sense that the variance in cell size remains bounded. Moreover, we compute the cell size variance as a function of model parameters applying the provided theorems and validate it by performing Monte Carlo simulations of the exact dynamics. In summary, our work provides an analytical approach for characterizing moments of a nonlinear stochastic dynamical system with broad applicability in studying random phenomena in both engineering and biology.