Multiplicative Periodic Signal Research Articles

Metastable oscillations in an evolutionary game: Synchronization and control

Evolutionary games with a finite number of players may manifest transient oscillatory dynamics before reaching an absorbing state. Such oscillations could be prone to synchronization, however its details and specific properties are not known. Here, we investigate synchronization of metastable oscillations in the finite-dimensional evolutionary game “Battle of the Sexes”, as well as the possibility of controlling the properties of these oscillations by a multiplicative periodic signal. The appropriately generalized frequency and phase quantifiers demonstrate the possibility of synchronization, and the asymmetry of frequency range, induced by the finite size of population. Evolution of oscillations under strengthening periodic driving depends on whether synchronization is reached: in the synchronous regime the amplitude of oscillations grows, while their lifetime decreases, whereas in the asynchronous regime their amplitude decays to zero with the quasi-stationary distribution becoming unimodal.

Impact of Non-Gaussian Noise and Time Delay on Stability and Stochastic Resonance for a FitzHugh-Nagumo Neural System Subjected to a Multiplicative Periodic Signal

In this paper, we focus on the investigations on the stochastic stability and the stochastic resonance (SR) phenomena for a FitzHugh-Nagumo system with time delay induced by a multiplicative non-Gaussian colored noise and an additive Gaussian colored noise. By use of the fast descent method, the unified colored noise approximation and the two-state theory for the SR, the stationary probability density function (SPDF) and the signal-to-noise ratio (SNR) caused by different noise terms and time delay are explored. The investigation results indicate that the two noise intensities, time delay and the departure parameter from the Gaussian noise can all reduce the probability density around the two stable states and destroy the stability of the neural system; while the two noise correlation times [Formula: see text] and [Formula: see text] can both improve the probability density around both stable states and reinforce the biological stability of the neural system. As regards the SNR, it is found that the two noise intensities and the departure coefficient can all weaken the SR effect, while time delay [Formula: see text] and the correlation time [Formula: see text] of the multiplicative noise will always magnify the SR phenomenon. It is worth to mention that the correlation time [Formula: see text] of the additive noise can stimulate the SR effect, but not alter the maximum of the SNR.

Stochastic delayed dynamics for a metapopulation system induced by non-Gaussian noise and a multiplicative periodic signal

In this paper, we focus on shedding light on the stochastic stability, the MFPT and stochastic resonance (SR) phenomena for a metapopulation system in the presence of time delay, the colored cross-correlated multiplicative non-Gaussian noise and the additive Gaussian colored noise. By use of the fast descent method and the approximation of the unified colored noise, the dynamical behaviors for the modified potential function and the signal-to-noise ratio (SNR) are analyzed in detail. It is discovered that the additive Gaussian colored noise, the noise correlation strength as well as its self-correlation time can all impair the stability of the population system and speed up the population decline. Conversely, the two self-correlation times of the external and internal noises can both consolidate the stability of the biological system and prolong the survival time of the population. With regard to the SNR for the metapopulation system caused by the noise terms and time delay, it is revealed that time delay, the noise correlation strength and its correlation time can all along weaken the SR phenomenon. On the contrary, two noise self-correlation times are able to amplify effectively the resonance effect. It is worth noting that the parameter of non-Gaussianity $$ q $$ in the SNR- $$ \tau $$ and SNR- $$ \lambda $$ plots can initially weaken the SR phenomena and then enhance the maximum of the SNR as it is further increased.

Stochastic Kinetics for a FitzHugh–Nagumo Neural System with Time Delay Driven by Non-Gaussian Noise and a Multiplicative Periodic Signal

In the present paper, the stability and the phenomena of stochastic resonance (SR) for a FitzHugh–Nagumo (FHN) system with time delay driven by a multiplicative non-Gaussian noise and an additive Gaussian white noise are investigated. By using the fast descent method, unified colored noise approximation and the two-state theory for the SR, the expressions for the stationary probability density function (SPDF) and the signal-to-noise ratio (SNR) are obtained. The research results show that the two noise intensities and time delay can always decrease the probability density at the two stable states and impair the stability of the neural system; while the noise correlation time [Formula: see text] can increase the probability density around both stable states and consolidate the stability of the neural system. Furthermore, the other noise correlation time [Formula: see text] can increase the probability at the resting state, but reduce that around the excited state. With respect to the SNR, it is discovered that the two noise strengths can both weaken the SR effect, while time delay [Formula: see text] and the departure parameter [Formula: see text] will always amplify the SR phenomenon. Moreover, the noise correlation time [Formula: see text] can motivate the SR effect, but not alter the peak value of the SNR. What’s most interesting is that the other noise correlation time [Formula: see text] can not only stimulate the SR phenomenon, but also results in the occurrence of two resonant peaks, whose heights are simultaneously improved because of the action of [Formula: see text].

Combined action of non-Gaussian noise and time delay on stochastic dynamical features for a metapopulation system driven by a multiplicative periodic signal

In this paper, we focus on investigating the steady state behaviors and the signal-to-noise ratio (SNR) for a stochastic metapopulation system with time delay induced by the cross-correlated multiplicative non-Gaussian noise and additive Gaussian noise. It is found by the numerical calculation that time delay, the strength of the multiplicative noise, the intensity of the additive one and the departure parameter from the Gaussian noise can all reduce greatly the stability of the biological population system and hold down the reproduction of the metapopulation, while the correlation time of the multiplicative noise and the noise correlation strength can strengthen the stability of the ecological system and facilitate the expansion of the population system. As regards the stochastic resonance effect (SR) evoked by time delay, the correlated noises and a multiplicative periodic signal, the consequence exhibits that due to the action of the noise correlation time τ, the SR effect is always enhanced. While time delay α, the departure parameter q and the noise intensity Q play mainly critical roles in weakening the SR and reducing the SNR. On different occasions, the noise correlation strength λ plays inverse roles in stimulating the SR effect. In particular, what is worth mentioning is that in the SNR-τ plots, time delay α, the noise intensities Q, M will all improve greatly the maximum of the SNR.

Impact of Time Delay and Non-Gaussian Noise on Stochastic Resonance and Stability for a Stochastic Metapopulation System Driven by a Multiplicative Periodic Signal

In the present paper, the stability of the population system and the phenomena of the stochastic resonance (SR) for a metapopulation system induced by the terms of time delay, the multiplicative non-Gaussian noise, the additive colored Gaussian noise and a multiplicative periodic signal are investigated in detail. By applying the fast descent method, the unified colored noise approximation and the SR theory, the expressions of the steady-state probability function and the SNR are derived. It is shown that multiplicative non-Gaussian noise, the additive Gaussian noise and time delay can all weaken the stability of the population system, and even result in population extinction. Conversely, the two noise correlation times can both strengthen the stability of the biological system and contribute to group survival. In regard to the SNR for the metapopulation system impacted by the noise terms and time delay, it is revealed that the correlation time of the multiplicative noise can improve effectively the SR effect, while time delay would all along restrain the SR phenomena. On the other hand, although the additive noise and its correlation time can stimulate easily the SR effect, they cannot change the maximum of the SNR. In addition, the departure parameter from the Gaussian noise and the multiplicative noise play the opposite roles in motivating the SR effect in different cases.

Time delay and non-Gaussian noise-induced stochastic stability and stochastic resonance for a metapopulation system subjected to a multiplicative periodic signal

In this paper, the stable state transformation and the effect of the stochastic resonance (SR) for a metapopulation system are investigated, which is disturbed by time delay, the multiplicative non-Gaussian noise, the additive colored Gaussian noise and a multiplicative periodic signal. By use of the fast descent method, the approximation of the unified colored noise and the SR theory, the dynamical behaviors for the steady-state probability function and the SNR are analyzed. It is found that non-Gaussian noise, the colored Gaussian noise and time delay can all reduce the stability of the biological system, and even lead to the population extinction. Inversely, the self-correlation times of two noises can both increase the stability of the population system and be in favor of the population reproduction. As regards the SNR for the metapopulation system induced by the noise terms and time delay, it is discovered that time delay and the correlation time of the multiplicative noise can effectively enhance the SR effect, while the multiplicative noise and the correlation time of the additive noise would all the time suppress the SR phenomena. In addition, the additive noise can effectively motivate the SR effect, but not alter the peak value of the SNR. It is worth noting that the departure parameter from the Gaussian noise plays the diametrical roles in stimulating the SR effect in different cases.

Impact of time delay and a multiplicative periodic signal on stochastic resonance and steady states shift for a stochastic insect outbreak system subjected to Gaussian noises

In this paper, we aim to investigate comprehensively the steady-states characteristics, the stochastic resonance phenomenon and the mean decline time for an insect outbreak system caused by the terms of the multiplicative, additive noises and time delay,. Our results exhibit that the multiplicative noise and the time delay can both reduce the stability of the biological system and speed up the extinction process of the insect population, while the additive noise can decrease the possibility of the decline of the biological population by a wide margin and make contribution to the survival and reproduction of the insect system to some extent. On the other hand, as regards to the stochastic resonance phenomenon (SR) induced by noise terms, time delay term and a weak multiplicative periodic signal, the numerical results show that the multiplicative noise intensity Q always suppresses the SR effect in any case, while the additive noise intensity M can inhibit the SR effect in the case of a big value of Q, but excite the maximum of the SNR for the case of a small value of Q. Moreover, time delay τ exerts mainly the inhibitory effect on the SR phenomenon except that in the SNR-Q plot.

Impact of colored cross-correlated noises on stochastic resonance and mean extinction rate for a metapopulation system with a multiplicative periodic signal

In this paper, we aim to explore the mean extinction rate and the phenomena of the stochastic resonance (SR) for a metapopulation system induced by a multiplicative periodic signal, colored cross-correlated multiplicative and additive Gaussian noises. By use of the fast descent method and the adiabatic approximation theory for the signal-to-noise ratio, we obtain the expression of the signal-to-noise ratio (SNR). Numerical results indicate that the various SR phenomena occur in the metapopulation system due to the variation of the noise terms and the correlation time. Specifically, the noise correlation always plays a critical role in motivating the SR phenomenon, while the multiplicative noise exerts the inhibition effect on the SR. Interestingly, the weak additive noise can stimulate the resonant peak of the SNR, while the further increase of the noise intensity will lead to the reduction of the SR effect. On the other hand, the noise correlation time τ plays antipodal roles in motivating the SR phenomenon under different circumstances. With regard to the mean extinction rate of the population from the boom state to the extinction one, by performing the numerical calculations, it is found that the additive noise always accelerate the extinction of the population, while the correlation noise will slow down the decline for the population. The role that the noise correlation time plays in the population extinction depends on the values that λ takes.

Stochastic resonance in a time-delayed tumor cell growth system driven by additive and multiplicative noises

In this paper, the stochastic resonance (SR) phenomenon in a time-delayed tumor cell growth system subjected to a multiplicative periodic signal, the multiplicative and additive noise is investigated. By applying the small time-delay method and two-state theory, the expressions of the mean first-passage time (MFPT) and signal-to-noise ratio (SNR) are obtained, then, the impacts of time delay, noise intensities and system parameters on the MFPT and SNR are discussed. Simulation results show that the multiplicative and additive noise always weaken the SR effect; while time delay plays a key role in motivating the SR phenomenon when noise intensities take a small value, it will restrain SR phenomenon when noise intensities take a large value; the cycle radiation amplitude always plays a positive role in stimulating the SR phenomenon, while, system parameters play different roles in motivating or suppressing SR under the different conditions of noise intensities.

Time delay effects of stochastic resonance induced by multiplicative periodic signal in the gene transcriptional regulatory model

Based on the universal transcriptional regulatory model proposed by Smolen et al.in 1998, we have studied the combined effects of cross-correlation noise, multiplicative periodic signal and time delay through modifying the ordinary differential equation with time evolution in gene transcriptional regulatory system. The expression of signal-to-noise ratio (SNR) is derived by applying the two-state theory in adiabatic limit and approximate analytical method for small time delay. The effects of multiplicative noise intensity, additive noise intensity, cross-correlation noise intensity and time delay on SNR are discussed by numerical calculation and graphical analysis. The results indicate that when the SNR is a function of multiplicative noise intensity, the peak value of the SNR rises as time delay increases; when SNR is a function of additive noise intensity, the value of SNR ascends as time delay increases; when SNR is a function of cross-correlation noise intensity, the peak value of SNR goes up with time delay increased. So when SNR is the function of multiplicative noise intensity, additive noise intensity and cross-correlation noise intensity respectively, time delay enhances the stochastic resonance.

Impact of colored cross-correlated non-Gaussian and Gaussian noises on stochastic resonance and stochastic stability for a metapopulation system driven by a multiplicative signal

In this paper, our aim is to investigate the steady state characteristics and the signal-to-noise ratio (SNR) for a stochastic metapopulation system including a multiplicative periodic signal caused by the terms of the colored cross-correlated multiplicative non-Gaussian noise and additive Gaussian noise. Numerical results indicate that the multiplicative noise, the additive one and the departure parameter from the Gaussian noise can all decrease the stability of the ecological population system and restrain the development of the metapopulation, while two noise correlation times and the strength of the noise correlation will enhance the stability of the biological system and promote the expansion of the population system. With regard to the stochastic resonance phenomenon (SR) induced by noise terms and a multiplicative weak periodic signal, the results illustrate that the noise correlation time τ and the strength of correlation noise λ can increase the SR effect greatly in most cases, while the intensity of the multiplicative noise Q mainly plays a part in suppressing the SR and weakening the SNR except that in the SNR-τ plot. Moreover, it is worth noting that the noise correlation time τ0 and the additive noise intensity M can play the diverse roles in enhancing or weakening the SR effect under the different system parameter conditions.

Colored Gaussian noises induced stochastic resonance and stability transition for an insect growth system driven by a multiplicative periodic signal

In this paper, we focus on investigating the steady-state shift behaviors and the stochastic resonance phenomenon (SR) for a biological insect population system with a multiplicative periodic signal caused by the terms of the colored multiplicative and additive noises. Our research results imply that the multiplicative noise and the self-correlation of the additive noise can weaken the stability of the biological system and restrain the growth of the insect population, while the additive noise and the self-correlation time of the multiplicative noise can strengthen the stability of the insect system and facilitate the biological population to breed. As regards to the phenomenon of the SR evoked by a multiplicative periodic signal, noise terms and their correlation times, the computed results show that the additive noise intensity M and the self- correlation time τ1 of the multiplicative noise can both improve the SR effect. Inversely, the multiplicative noise intensity Q and the self-correlation time τ2 of the additive noise can suppress together the SR phenomenon. Whereas, it should be pointed out that in the SNR-Q and SNR-M plots, the two self-correlation times can both motivate a resonant peak, but not change the peak value of the SNR no matter how the two noise correlation times vary.

Double time-delays induced stochastic dynamical characteristics for a metapopulation system subjected to the associated noises and a multiplicative periodic signal

In this paper, we aim to investigating in detail the stability of the system, the mean extinction time and stochastic resonance (SR) phenomena caused by a multiplicative periodic signal for a dual time-delayed metapopulation system subjected to cross-correlated noises. By use of the fast descent method, the small time delay approximation method and the SR theory, we obtain the expressions of the steady state probability distribution function, the mean first-passage time and signal-to-noise ratio (SNR). Numerical results indicate that the multiplicative, additive and association noises together with time delay τ can all accelerate the transition from the stable state of big density to the extinction one and play significant roles in weakening the stability and shortening the mean extinction time of the metapopulation. In particular, the additive noise and time delay τ can result in the crash of the system, while another time delay θ can strengthen the biological system stability and extend the declining time for the population. On the other hand, with respect to the SNR, the figures show that time delay τ plays entirely antipodal roles in motivating stochastic resonance (SR) in a variety of different situations. Conversely, the multiplicative noise intensity Q and time delay θ all along produce negative effect on exciting the SR. Meanwhile, the increase of the weak additive noise intensity M can stimulate the SR phenomenon, but the bigger values of M will suppress the SNR and SR phenomenon. The strength of the noise correlation λ plays an important role in restraining the SR in most cases except that it does in the plot of SNR-Q.

Stability and Stochastic Resonance for a Time-Delayed Cancer Development System Subjected to Noises and Multiplicative Periodic Signal

In this paper, the stability and the phenomenon of stochastic resonance (SR) for a stochastic time-delayed cancer development system that is induced by the multiplicative periodic signal, the multiplicative and the additive noises are investigated. By using the fast descent method, small time delay method and the two-state theory, the expressions of the steady state probability distribution function and the signal-to-noise ratio (SNR) are obtained. Numerical results reflect that the multiplicative and additive noise always restrain the diffusion of the cancer cells. Whereas, the time delay can not only control the spread of the tumor cells, but also suppress the extinction of cancer cells. Meanwhile, the conventional SR occurs in the tumor cell growth model under the excitation of different noises and time delay. In conclusion, the multiplicative noise always plays a critical role in restraining SR, a smaller additive noise can stimulate the SR, but the larger additive noise can weaken the SR and SNR. In particular, the time delay displays relatively complicated effects on the SR phenomenon of the system. It plays different roles in motivating or suppressing SR under the different conditions of parameters.

Stochastic resonance for a forest growth system subjected to non-Gaussian noises and a multiplicative periodic signal

In this paper, the stochastic resonance (SR) phenomenon induced by a multiplicative periodic signal in a forest growth system which is excited by additive Gaussian noise, multiplicative non-Gaussian noise and noise correlation time is investigated. According to the fast descent method, the unified colored noise approximation and the SR theory, the analytical expression of the signal-to-noise ratio (SNR)is derived in the adiabatic limit. Via numerical calculations, each effect of the addictive noise intensity, the multiplicative noise intensity and the departure parameter from the Gaussian noise upon the signal-to-noise ratio (SNR) is discussed. It is shown that the addictive noise intensity always plays a significant role in stimulating the SR phenomenon. Conversely, the multiplicative noise intensity generally plays a remarkable role in restraining the effect of SR in most cases except for the case of M=0.01. Moreover, the noise correlation time can also excite the SR effect. On the other hand, the departure parameter from the Gaussian noise and the forest growth rate can produce entirely different effects on the SNR under the different conditions of the system parameters.

Stochastic resonance and stability for an ecological vegetation growth system driven by colored noises and multiplicative signal

In this paper, we investigate the steady-state properties and the transition rate for an ecological vegetation growth system induced by the terms of the colored multiplicative and additive noises. Numerical results indicate that the multiplicative noise and the additive one can reduce the stability of the ecological system and slow down the development velocity of the vegetation, while two noise self-correlation times can increase the stability of the system and speed up the expansion process of the vegetation system. With respect to the stochastic resonance (SR) phenomenon caused by noise terms and a multiplicative weak periodic signal, the results show that the additive noise always enhances the SR effect, two noise self-correlation time terms can produce SR phenomenon, but play opposite roles in enhancing or inhibiting the SR effect under different parameter conditions. In particular, the two self-correlation times can keep up the maximum of the signal-to-noise ratio (SNR) invariant in specific situations. Analogously, the multiplicative noise can not only improve the SNR, but also restrain the SR phenomenon in different cases.

Multiple absolute negative mobilities.

In this paper, we investigate transport of an inertial particle in a spatially symmetric potential and subjected to two harmonic signals with different frequencies in both deterministic and stochastic cases. Numerical results indicate that: (i) In the deterministic case, the two harmonic signals can induce many (up to six) segments of negative slopes in the curve of mean velocity vs. external constant force, i.e., a multiple absolute negative mobilities (ANMs) effect. But the occurrence of the effect depends on their frequencies and amplitudes. (ii) For the stochastic case, the multiple ANMs relay on stable index and symmetry parameter of the Lévy noise. In the case of symmetric noise, appropriate stable index makes the multiple ANMs be the strongest. Our further investigations indicate that an indispensable condition for the multiple ANMs to occur in the system is the temporal symmetry breaking by one multiplicative periodic signal and one additive periodic signal.

Stochastic resonance and stability for a stochastic metapopulation system subjected to non-Gaussian noise and multiplicative periodic signal

In this paper, the stability and stochastic resonance (SR) phenomenon induced by the multiplicative periodic signal for a metapopulation system driven by the additive Gaussian noise, multiplicative non-Gaussian noise and noise correlation time is investigated. By using the fast descent method, unified colored noise approximation and McNamara and Wiesenfeld’s SR theory, the analytical expressions of the stationary probability distribution function and signal-to-noise ratio (SNR) are derived in the adiabatic limit. Via numerical calculations, each effect of the addictive noise intensity, the multiplicative noise intensity and the correlation time upon the steady state probability distribution function and the SNR is discussed, respectively. It is shown that multiplicative, additive noises and the departure parameter from the Gaussian noise can all destroy the stability of the population system. However, the noise correlation time can consolidate the stability of the system. On the other hand, the correlation time always plays an important role in motivating the SR and enhancing the SNR. Under different parameter conditions of the system, the multiplicative, additive noises and the departure parameter can not only excite SR phenomenon, but also restrain the SR phenomenon, which demonstrates the complexity of different noises upon the nonlinear system.

Effect of Time Delay and Cross-Correlated Noises on Stochastic Resonance for a Metapopulation System with a Multiplicative Periodic Signal

In this paper, stochastic resonance (SR) in a time-delayed metapopulation system subjected to cross-correlated noises and a multiplicative signal is investigated. By using the fast descent method and the two-state theory, we obtain the expression of the signal-to-noise ratio (SNR). Numerical results show that the conventional SR phenomenon appears in the metapopulation model under different values of the system parameters. The results demonstrate that the strength of the correlation noise λ always plays a decisive role in motivating the SR phenomenon. On the other hand, for the case of λ ＞ 0, the intensities of the multiplicative and additive noises and the time delay can all weaken the effect of SR; for the case of λ ＜ 0, the multiplicative noise can restrain the effect of SR, the weak additive noise can excite the effect of SR, the time delay τ can produce different effects on the SNR under the conditions of different parameters.