@article{EngelOliconMendezWehlitzetal., author = {Engel, Maximilian and Olic{\´o}n-M{\´e}ndez, Guillermo and Wehlitz, Nathalie and Winkelmann, Stefanie}, title = {Synchronization and random attractors in reaction jump processes}, series = {Journal of Dynamics and Differential Equations}, journal = {Journal of Dynamics and Differential Equations}, doi = {10.1007/s10884-023-10345-4}, abstract = {This work explores a synchronization-like phenomenon induced by common noise for continuous-time Markov jump processes given by chemical reaction networks. Based on Gillespie's stochastic simulation algorithm, a corresponding random dynamical system is formulated in a two-step procedure, at first for the states of the embedded discrete-time Markov chain and then for the augmented Markov chain including random jump times. We uncover a time-shifted synchronization in the sense that—after some initial waiting time—one trajectory exactly replicates another one with a certain time delay. Whether or not such a synchronization behavior occurs depends on the combination of the initial states. We prove this partial time-shifted synchronization for the special setting of a birth-death process by analyzing the corresponding two-point motion of the embedded Markov chain and determine the structure of the associated random attractor. In this context, we also provide general results on existence and form of random attractors for discrete-time, discrete-space random dynamical systems.}, language = {en} } @article{WehlitzSadeghiMontefuscoetal., author = {Wehlitz, Nathalie and Sadeghi, Mohsen and Montefusco, Alberto and Sch{\"u}tte, Christof and Pavliotis, Grigorios A. and Winkelmann, Stefanie}, title = {Approximating particle-based clustering dynamics by stochastic PDEs}, abstract = {This work proposes stochastic partial differential equations (SPDEs) as a practical tool to replicate clustering effects of more detailed particle-based dynamics. Inspired by membrane mediated receptor dynamics on cell surfaces, we formulate a stochastic particle-based model for diffusion and pairwise interaction of particles, leading to intriguing clustering phenomena. Employing numerical simulation and cluster detection methods, we explore the approximation of the particle-based clustering dynamics through mean-field approaches. We find that SPDEs successfully reproduce spatiotemporal clustering dynamics, not only in the initial cluster formation period, but also on longer time scales where the successive merging of clusters cannot be tracked by deterministic mean-field models. The computational efficiency of the SPDE approach allows us to generate extensive statistical data for parameter estimation in a simpler model that uses a Markov jump process to capture the temporal evolution of the cluster number.}, language = {en} }