Monostable Systems Research Articles

Bistable travelling waves for a reaction and diffusion model with seasonal succession**Research supported in part by the NSERC of Canada and the MITACS of Canada.

This paper is devoted to the study of a periodic reaction–diffusion competition model, which describes the propagation of two competitive species in bad and good seasons. The existence and global stability of time-periodic bistable travelling waves are established for such a system under appropriate conditions. The methods involve the upper and lower solutions, spreading speeds of monostable systems, and the monotone semiflows approach.

A study of the accuracy of moment-closure approximations for stochastic chemical kinetics

Moment-closure approximations have in recent years become a popular means to estimate the mean concentrations and the variances and covariances of the concentration fluctuations of species involved in stochastic chemical reactions, such as those inside cells. The typical assumption behind these methods is that all cumulants of the probability distribution function solution of the chemical master equation which are higher than a certain order are negligibly small and hence can be set to zero. These approximations are ad hoc and hence the reliability of the predictions of these class of methods is presently unclear. In this article, we study the accuracy of the two moment approximation (2MA) (third and higher order cumulants are zero) and of the three moment approximation (3MA) (fourth and higher order cumulants are zero) for chemical systems which are monostable and composed of unimolecular and bimolecular reactions. We use the system-size expansion, a systematic method of solving the chemical master equation for monostable reaction systems, to calculate in the limit of large reaction volumes, the first- and second-order corrections to the mean concentration prediction of the rate equations and the first-order correction to the variance and covariance predictions of the linear-noise approximation. We also compute these corrections using the 2MA and the 3MA. Comparison of the latter results with those of the system-size expansion shows that: (i) the 2MA accurately captures the first-order correction to the rate equations but its first-order correction to the linear-noise approximation exhibits the wrong dependence on the rate constants. (ii) the 3MA accurately captures the first- and second-order corrections to the rate equation predictions and the first-order correction to the linear-noise approximation. Hence while both the 2MA and the 3MA are more accurate than the rate equations, only the 3MA is more accurate than the linear-noise approximation across all of parameter space. The analytical results are numerically validated for dimerization and enzyme-catalyzed reactions.

Bistable travelling waves in competitive recursion systems

This paper is devoted to the study of spatial dynamics for a class of discrete-time recursion systems, which describes the spatial propagation of two competitive invaders. The existence and global stability of bistable travelling waves are established for such systems under appropriate conditions. The methods involve the upper and lower solutions, spreading speeds of monostable systems, and the monotone semiflows approach.

A new perspective on stochastic resonance in monostable systems

Stochastic resonance induced by multiplicative white noise is theoretically studied in forced damped monostable oscillators. A stochastic amplitude equation is derived for the oscillation envelope, which has a linear stochastic resonance. This phenomenon is persistent when nonlinearities are considered. We propose three simple systems—a horizontally driven pendulum, a forced electrical circuit and a laser with an injected signal—that display this stochastic resonance.

Spreading speeds and traveling waves for abstract monostable evolution systems

This paper is devoted to the development of the theory of spreading speeds and traveling waves for abstract monostable evolution systems with spatial structure. Under appropriate assumptions, we show that the spreading speeds coincide with the minimal wave speeds for monotone traveling waves in the positive and negative directions. Then we use this theory to study the spatial dynamics of a parabolic equation in a periodic cylinder with the Dirichlet boundary condition, a reaction–diffusion model with a quiescent stage, a porous medium equation in a tube, and a lattice system in a periodic habitat.

Stochastic resonance and antiresonance in monostable systems

Faculty of Radiophysics, University of Nizhny Novgorod,23 Gagarin Ave, Nizhny Novgorod, 603950, Russia(Dated: August 14, 2008)We show, the SR can appear in monostable overdamped systems driven by additive mix of peri-odical signal and white Gaussian noise. It can be observed as non-monotonic dependence of SNR onthe input noise intensity. In this sense it is similar to classical SR observed in overdamped bistablesystems with potential barrier.

INVESTIGATION OF STOCHASTIC RESONANCE IN MONOSTABLE SYSTEMS

The phenomena of stochastic resonance is studied in overdamped nonlinear monostable systems driven by a periodic signal and Gaussian white noise. It is shown that the signal power amplification as a function of input noise intensity can be different depending on nonlinearity: it can monotonically grow, decrease and it can reach a maximum at certain value of the noise intensity. Nevertheless, the output signal to noise ratio is shown to be always a decreasing function of input noise intensity.

Existence of additional unstable equilibria for some monostable systems

We prove the existence of additional unstable equilibria E ± for a class of “monostable” reaction–diffusion systems u t = A u x x + f ( u ) , u ( x , t ) ∈ R N , having known equilibria S < T with S “stable” and T “unstable” such that the derivative d f [ T ] has an even number of unstable eigenvalues. The proofs use positive-operator theory and degree theory. Systems in the class considered are known to admit monotone increasing travelling-wave solutions w ( x − c t ) connecting ordered stable and unstable equilibria for all c greater than or equal to a positive critical value and the existence of E ± > S yields two one-parameter families of travelling waves connecting S to E ± in addition to the family connecting S to T .

Dynamical bimodality in equilibrium monostable systems

General features of the stochastic dynamics of classical systems approaching a thermodynamic equilibrium Gibbs state are studied via the numerical analysis of time-dependent solutions of the Fokker-Planck equation for an overdamped particle in various monostable potentials. A large class of initial states can dynamically bifurcate during its time evolution into bimodal transient states, which in turn wear off when approaching the long-time regime. Suitable quantifiers characterizing this transient dynamical bimodality, such as its lifetime, the positions of maxima, and the time-dependent well depth of the probability distribution, are analyzed. Some potential applications are pointed out that make use of this interesting principle which is based on an appropriately chosen initial preparation procedure.

Signal-to-noise ratio gain by stochastic resonance in a bistable system

It was shown recently that the signal-to-noise ratio (SNR) could be improved by stochastic resonance (SR) in certain monostable systems and certain systems with monotonous nonlinearity working in the nonlinear response (NLR) regime. Here we demonstrate that a simple bistable system driven by periodic pulse train and band-limited Gaussian white noise can also provide an SNR gain. We show the reasons why SNR gain had not been found in double well potential systems, despite strong efforts made during the last 15 years.

Transient bimodality of nonequilibrium states in monostable systems with noise

We discuss a new type of monostable potential profiles for which the phenomenon of transient bimodality occurs in dynamic systems. It is shown that transient bimodality occurs if the potential profile describing the dynamic system has a linear part. A new method for calculating the lifetime of the transient bimodality is proposed. We show that the lifetime of transient bimodality for the considered potential profiles increases in proportion to the fluctuation intensity.

Travelling–wave solutions to monostable reaction–diffusion systems of mixed monotone type

Theorems concerning the existence and stability of travelling–wave solutions to mono–stable reaction–diffusion systems of mixed monotone type are established. These results are applied to an autocatalytic system, which is a particular case in the class of monostable systems of mixed monotone type. A series of numerical experiments is conducted to simulate formation and propagation of waves.

Boundary effects on the dynamics of monostable reaction-diffusion systems

We study a monostable reaction-diffusion model in a bounded domain, subjected to partially reflecting boundary conditions. We analyze the stability of the arising patterns and detect a bifurcation of the uniform solution induced by changes in the reflectivity of the boundaries. We examine the critical slowing down of the system's dynamics in the neighborhood of the bifurcation point by analyzing its non-equilibrium potential.

Stochastic resonance at higher harmonics in monostable systems

The nonlinear response of a system subject to a periodic force in the presence of noise is investigated. It is shown that a physical system of a general type manifests stochastic resonance at higher harmonics. The stochastic resonance curves are calculated for several monostable systems. Higher harmonic stochastic resonance is measured on a monostable part of a domain wall pinned by a magnetic defect in a ferrite-garnet film.

Global analysis of pattern selection and bifurcations in monostable reaction-diffusion systems

We study a piecewise linear version of a one-component monostable reaction diffusion model in a bounded domain, subjected to partially reflecting boundary conditions (“ albedo” b.c.). We analyze the local and the global stability of the merging patterns and detect a bifurcation of the uniform solution induced by changes in the reflectivity of the boundaries.

Divergent Signal-to-Noise Ratio and Stochastic Resonance in Monostable Systems.

We present a class of systems for which the signal-to-noise ratio always increases when increasing the noise and diverges at infinite noise level. This new phenomenon is a direct consequence of the existence of a scaling law for the signal-to-noise ratio and implies the appearance of stochastic resonance in some monostable systems. We outline applications of our results to a wide variety of systems pertaining to different scientific areas. Two particular examples are discussed in detail.

Stochastic resonance in monostable systems

The first observations of noise-induced enhancements and phase shifts of a weak periodic signal-characteristics signatures of stochastic resonance (SR)-are reported for a monostable system. The results are shown to be in good agreement with a theoretical description based on linear-response theory and the fluctuation dissipation theorem. It is argued that SR is a general phenomenon that can in principle occur for any underdamped nonlinear oscillator.

Nonconventional stochastic resonance

It is argued, on the basis of linear response theory (LRT), that new types of stochastic resonance (SR) are to be anticipated in diverse systems, quite different from the one most commonly studied to date, which has a static double-well potential and is driven by a net force equal to the sum of periodic and stochastic terms. On this basis, three new nonconventional forms of SR are predicted, sought, found, and investigated both theoretically and by analogue electronic experiment: (a) in monostable systems; (b) in bistable systems with periodically modulated noise; and (c) in a system with coexisting periodic attractors. In each case, it is shown that LRT can provide a good quantitative description of the experimental results for sufficiently weak driving fields. It is concluded that SR is a much more general phenomenon than has hitherto been appreciated.

Power spectra of noise-driven nonlinear systems and stochastic resonance

The results of recent experimental and theoretical investigations of the spectral densities of fluctuations (SDFs) of noise-driven nonlinear dynamical systems are reviewed. Emphasis is placed on the analysis of the shapes and intensities of peaks in the SDFs. Three different types of phenomena are considered. First, the SDFs of a class of monostable underdamped nonlinear systems, in which the variation of eigenfrequency with energy is nonmonotonic, are investigated. It is shown that they exhibit zero-dispersion peaks and noise-induced spectral narrowing, as well as zero-frequency peaks. Secondly, it is demonstrated that systems bistable in an external periodic field can exhibit supernarrow spectral peaks within the range of a kinetic phase transition. Finally, recent results in stochastic resonance (SR) are reviewed, including phase shifts, giant nonlinearities for weak noise, SR for periodically modulated noise intensity, and high-frequency SR for periodic attractors.

A.c. or d.c. imposed on enzyme systems : Monitoring of monostable systems and memory in multistable systems

The construction of a theory of enzyme diffusion-electromigration-reaction regimes has been attempted. An electric field acts on the ionic species present in the membrane; when these species are proton, substrate, activator or inhibitor, they constitute the mediators of the action of the electric field on the enzyme membrane activity. Depending on the reaction scheme and the values of the different parameters (conductivity level, model structure, existence or non-existence of multiple steady states), various results have been obtained: continuous effects (enzyme membrane activity modulations) with monostable systems ( a.c. and d.c.), and discontinuous ones (hysteresis and memory induction) have been pointed out.