Intrinsic noise induced resonance in presence of sub-threshold signal in Brusselator

In a system of non-linear chemical reactions called the Brusselator, we show that {\it intrinsic noise} can be regulated to drive it to exhibit resonance in the presence of a sub-threshold signal. The phenomena of periodic stochastic resonance and aperiodic stochastic resonance, hitherto studied mostly with extrinsic noise, is demonstrated here to occur with inherent systemic noise using exact stochastic simulation algorithm due to Gillespie. The role of intrinsic noise in a couple of other phenomena is also discussed.Comment: 7 pages, 5 figure

Noise-induced instabilities in a stochastic Brusselator

We consider a stochastic version of the so-called Brusselator - a mathematical model for a two-dimensional chemical reaction network - in which one of its parameters is assumed to vary randomly. It has been suggested via numerical explorations that the system exhibits noise-induced synchronization when time goes to infinity. Complementing this perspective, in this work we explore some of its finite-time features from a random dynamical systems perspective. In particular, we focus on the deviations that orbits of neighboring initial conditions exhibit under the influence of the same noise realization. For this, we explore its local instabilities via finite-time Lyapunov exponents. Furthermore, we present the stochastic Brusselator as a fast-slow system in the case that one of the parameters is much larger than the other one. In this framework, an apparent mechanism for generating the stochastic instabilities is revealed, being associated to the transition between the slow and fast regimes

Mesa-type patterns in the one-dimensional Brusselator and their stability

The Brusselator is a generic reaction-diffusion model for a tri-molecular chemical reaction. We consider the case when the input and output reactions are slow. In this limit, we show the existence of $K$-periodic, spatially bi-stable structures, \emph{mesas}, and study their stability. Using singular perturbation techniques, we find a threshold for the stability of $K$ mesas. This threshold occurs in the regime where the exponentially small tails of the localized structures start to interact. By comparing our results with Turing analysis, we show that in the generic case, a Turing instability is followed by a slow coarsening process whereby logarithmically many mesas are annihilated before the system reaches a steady equilibrium state. We also study a ``breather''-type instability of a mesa, which occurs due to a Hopf bifurcation. Full numerical simulations are shown to confirm the analytical results.Comment: to appear, Physica

Multi-Scale Dynamical Analysis on Circadian Rhythms of \u3cem\u3eArabidopsis thaliana\u3c/em\u3e

Adapting to the 24-hour periodic environment on the Earth, plants have evolved sets of chemical reactions that regulate their circadian rhythms. A number of research groups studying these circadian reactions in the common laboratory plant Arabidopsis thaliana have developed eleven, increasingly elaborate, chemical kinetic models based on genetic feedback loops. Each model consists of a system of coupled nonlinear ordinary differential equations. We find these models are all situated near a Hopf bifurcation in parameter space. This suggests that there may be some biological significance corresponding to this mathematical property. To study the properties of these systems related to the Hopf bifurcation, we first numerically compute the solutions to the kinetic models for Arabidopsis thaliana. At the whole plant scale, we perform a weakly nonlinear analysis, the Reductive Perturbation Method, on each model near bifurcation to predict the amplitude and frequency of the oscillating concentration of chemical species from the Stuart-Landau amplitude equation. By scaling the numerical frequencies and amplitudes by our theoretical predictions, we show that the solutions to all these models collapse into a universal parameter-free form. Then, we implement Gillespie’s Stochastic Simulation Algorithm to simulate the system at the single-cell level and account for random fluctuations in molecule numbers. We relate the two approaches and discuss some implications of our results for improving future modeling efforts to ensure that the models are consistent with each other and with the dynamics of the Arabidopsis thaliana circadian rhythms. Finally, we comment on the possible biological significance of the models\u27 mathematical features

Stochastic sensitivity and noise-induced bifurcations of limit cycles

Deformations of limit cycles for dynamical systems forced by random disturbances are studied. A mathematical tool based on stochastic sensitivity analysis is shortly presented. A phenomenon of noise-induced bifurcation of supersensitive limit cycle of randomly forced Brusselator is discussed

Effect of a shot-noise generator on oscillations in a Salnikov model of an exothermal chemical reaction

Our purpose is to study the impact of chaos-generators on the dynamics of nonlinear systems. As an example two simplified models of chemical reactions have been chosen with different coupling to the external noise. Numerical analysis together with analytic predictions for a stationary situation show possibility of noise induced periodicity