Anticipated synchronization in stochastic individual-based models

Abstract

[eng] In this work we study anticipated synchronisation in stochastic individual-based models. Anticipated synchronisation is a special type of synchronisation in which instead of having all the systems running with the same phase one system anticipates the behaviour of the other. It has attracted the attention of many researchers and has been well studied for dynamical systems but it has not been studied before for stochastic individual-based models. The model used consists of two identical copies of a Brusselator. The first copy does not have any coupling (master system). The second copy has to coupling terms that are multiplied by the same coupling strength. The first coupling is to the current state of the master system. The second coupling is to its own state at a time equal to the current time minus a given time delay (slave system). This is a typical configuration to observe anticipated synchronisation. We will study two regions for the parameters of the Brusselator, in the first one the deterministic system have oscillations and in the second one the deterministic system goes to a fixed point and the stochastic model has noise-induced oscillations. For the first region of parameters values the stochastic and the deterministic models have oscillations. For the second region of parameters values however the stochastic systems has oscillations while the deterministic system just goes to a fixed point. In this second region we performed the theoretical analysis. In order to study it theoretically we need a frame to work with time delayed reactions in stochastic processes. After the theoretical frame is clear we need to carry out a Kramers-Moyal expansion of the master equation and a linear-noise approximation of the resulting Langevin equation. Within these two approximations we are able to obtain an analytical expression for the power spectra and for the cross-spectral density. Then we can compute from these two functions the auto-correlation function and the crosscorrelation function. The cross-correlation will be very important in order to determine if the system exhibits anticipated synchronisation since if we have synchronisation between two oscillating system the cross-correlation will be a periodic function and the position of the first maximum will give the time shift between the two signals. We also run simulations in order to check the theoretical results. For the simulations we use a modification of the Gillespie algorithm that allow us to introduce time delayed reactions. First of all, we study the region of parameters values with oscillations in the deterministic system. We see that in this region the deterministic and the stochastic models have indeed anticipated synchronisation for a given region of values of the coupling strength and the time delay. We give a conservative estimation of the region with anticipated synchronisation for both models. The regions with anticipated synchronisation are as expected similar in the deterministic and the stochastic models and the only differences can be attributed to finite size effects. Finally, we study the region of parameters values of the Brusselator with noise-induced oscillations. We checked that the theoretical results are quite similar to the ones obtained from the simulations. We also observed anticipated synchronisation in this region of parameters values.