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.     

Delay-induced state transition and resonance in periodically driven tumor model with immune surveillance

The phenomenon of stochastic resonance (SR) in a tumor growth model under the presence of immune surveillance is investigated. Time delay and cross-correlation between multiplicative and additive noises are considered in the system. The signal-to-noise ratio (SNR) is calculated when periodic signal is introduced multiplicatively. Our results show that: (i) the time delay can accelerate the transition from the state of stable tumor to that of extinction, however the correlation between two noises can accelerate the transition from the state of extinction to that of stable tumor; (ii) the time delay and correlation between two noises can lead to a transition between SR and double SR in the curve of SNR as a function of additive noise intensity, however for the curve of SNR as a function of multiplicative noise intensity, the time delay can cause the SR phenomenon to disappear, and the cross-correlation between two noises can lead to a transition from SR to stochastic reverse-resonance. Finally, we compare the SR phenomenon for the multiplicative periodic signal with that for additive periodic signal in the tumor growth model with immune surveillance.     

Influences of noise correlation and time delay on stochastic resonance induced by multiplicative signal in a cancer growth system

This paper investigates the stochastic resonance (SR) phenomenon induced by the multiplicative periodic signal in a cancer growth system with the cross-correlated noises and time delay. To describe the periodic change of the birth rate due to the periodic treatment, a multiplicative periodic signal is added to the system. Under the condition of small delay time, the analytical expression of the signal-to-noise ratio R_{\rm SNR} is derived in the adiabatic limit. By numerical calculation, the effects of the cross-correlation strength \lambda and the delay time \tau on R_{\rm SNR} are respectively discussed. The existence of a peak in the curves of R_{\rm SNR} as a function of the noise intensities indicates the occurrence of the SR phenomenon. It is found that \lambda and \tau play opposite role on the SR phenomenon, i.e., the SR is suppressed by increasing \lambda whereas it is enhanced with the increase of \tau, which is different from the case where the periodic signal is additive.     

Delay induced steady-state transition and stochastic resonance for an ecological vegetation growth system subjected to multiplicative and additive noises

In this paper, our aim is to investigate the steady state properties and stochastic resonance (SR) phenomenon for an ecological vegetation growth system with time delay induced by the multiplicative and additive noises. Numerical results show that the SR phenomenon caused by time delay, different noise terms and a weak periodic signal occurs in the vegetation growth model under different values of system parameters. With regard to the stationary state properties of the vegetation system, the results indicate that the terms of different noises and time delay can all accelerate the shift from the substantial state to the barren one of the ecological system, restrain the development of the vegetation system and weaken the stability of the ecological system. On the other hand, the additive noise strength always enhances the SNR and the SR phenomenon, while the intensity of multiplicative noise often reduces the effect of the SR. In particular, time delay can play different roles in exciting the SR phenomenon 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.     

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.     

Parameter-induced stochastic resonance in an over-damped linear system

Stochastic resonance (SR) in an over-damped linear system subjected to an excitation of bias signal modulated noise with multiplicative and additive noises is investigated. We obtain the exact expressions of the first two moments and the signal-to-noise ratio (SNR) of the output by using linear-response theory. The SNR depends non-monotonically on the intensity and the correlation time of multiplicative noise, the correlation time of additive noise, the intensity of the cross noise between multiplicative and additive noise, as well as the external field frequency. The conventional SR, the SR in a broad sense and the bona fide SR are found in the system. The influences of the asymmetries of multiplicative and additive noise, the correlation rate of the cross noise, the intensity of additive noise, the amplitude of signal and the bias on the SNR are analyzed. Moreover, we pointed out that SR can be realized by tuning the system parameter with fixed noise, i.e., parameter-induced stochastic resonance (PSR) exists.     

关联噪声驱动的非对称双稳系统的随机共振

运用统一色噪声近似理论和两态模型理论,研究了周期矩形信号和关联的乘性色噪声和加性白噪声驱动的非对称双稳系统的随机共振现象,得到了适合信号任意幅值的信噪比表达式.信噪比是乘性噪声强度、加性噪声强度、乘性噪声自关联时间、噪声耦合强度的非单调函数,所以该双稳系统中出现了随机共振.同时,调节加性噪声强度比调节乘性噪声强度更容易产生随机共振.势阱静态非对称性和噪声之间的耦合强度对信噪比的影响是不同的. 关键词： 非对称双稳系统 随机共振 信噪比 周期矩形信号     

光学双稳系统中的随机共振

运用绝热近似理论，研究了由加性噪声和乘性噪声及周期信号驱动的光学双稳系统的随机共振现象. 发现该模型中输出信噪比R～随着加性噪声强度D_a的变化曲线中会出现随机共振现象，而信噪比R～随着乘性噪声强度D_m的变化曲线是单调减小的，信噪比曲线中没有出现随机共振现象. 因此，加性噪声和乘性噪声对输出信噪比的影响是不同的. 关键词： 随机共振 信噪比 乘性噪声 加性噪声     

双稳态系统中随机共振和相干共振的相关性

研究了加性噪声和乘性噪声共同驱动的双稳态系统中的随机共振和相干共振现象. 针对加性噪声和乘性噪声之间不存在关联性和存在关联性两种情形, 引入一种适当的能同时表征随机共振和相干共振的指标, 应用一阶欧拉方法, 通过数值模拟对系统的随机共振和相干共振现象进行研究. 结果表明在弱噪声驱动下, 随着加性噪声强度的增加, 当系统出现相干共振时, 如果给系统外加一个弱周期驱动力, 几乎在同一时刻, 系统也出现了随机共振现象; 但随着乘性噪声强度的增加, 仅当加性噪声和乘性噪声之间相关时, 此结论成立. 并且系统参数对相干共振和随机共振的影响是一致的. 关键词： 双稳态系统 随机共振 相干共振     

由加性和乘性噪声驱动的非对称双稳系统的随机共振

根据信噪比理论，研究了由乘性和加性白噪声以及周期矩形信号共同作用的非对称双稳系统的随机共振。推导出了信噪比的解析表达式，并且该表达式适用于任意的信号振幅。数值分析表明乘性噪声强度D和加性噪声强度α对信噪比的影响是不同的：对应于任意的一个非对称系数r的值，SNR-α 曲线比SNR-D曲线更容易出现随机共振。即当系统的双阱不是很不对称的时候，改变加性噪声比改变乘性噪声更容易产生随机共振。此外，势井的非对称性能够减小信噪比。     

Effect of time delay in FitzHugh-Nagumo neural model with correlations between multiplicative and additive noises

We study the effect of time delay in the FitzHugh-Nagumo neural model with correlations between multiplicative and additive noise terms. Based on the corresponding Fokker-Planck equation, the explicit expressions of the stationary probability distribution function (SPDF), the mean first passage time (MFPT) and the signal-to-noise ratio (SNR) are obtained, respectively. Research results show that: (i) the system undergoes a succession of two phase transitions (i.e., the reentrance phenomenon) as the noise correlation parameter is increased and a (single) phase transition as the time delay is increased. (ii) The MFPT as a function of the multiplicative noise intensity exhibits a maximum. This maximum for MFPT identifies the noise enhanced stability (NES) effect, the noise correlation parameter intensifies the NES effect while the time delay, and the additive noise intensity weakens it. (iii) The existence of a maximum in the SNR as a function of the multiplicative noise intensity is the identifying characteristic of the stochastic resonance (SR) phenomenon, the noise correlation parameter enhances the SR while the time delay, and the additive noise intensity weaken it.     

Correlated noise-based switches and stochastic resonance in a bistable genetic regulation system

The correlated noise-based switches and stochastic resonance are investigated in a bistable single gene switching system driven by an additive noise (environmental fluctuations), a multiplicative noise (fluctuations of the degradation rate). The correlation between the two noise sources originates from on the lysis-lysogeny pathway system of the λ phage. The steady state probability distribution is obtained by solving the time-independent Fokker-Planck equation, and the effects of noises are analyzed. The effects of noises on the switching time between the two stable states (mean first passage time) is investigated by the numerical simulation. The stochastic resonance phenomenon is analyzed by the power amplification factor. The results show that the multiplicative noise can induce the switching from “on” → “off” of the protein production, while the additive noise and the correlation between the noise sources can induce the inverse switching “off” → “on”. A nonmonotonic behaviour of the average switching time versus the multiplicative noise intensity, for different cross-correlation and additive noise intensities, is observed in the genetic system. There exist optimal values of the additive noise, multiplicative noise and cross-correlation intensities for which the weak signal can be optimal amplified.     

色交叉关联噪声作用下集合种群的稳定性和平均灭绝时间

在Levins模型的基础上研究了色交叉关联噪声对集合种群稳定性的影响, 应用Fokker-Plank方程得到了系统的稳态概率密度函数, 运用最快下降法得到了平均灭绝时间的解析式. 结果表明: 两噪声色关联时, 加性噪声强度和乘性噪声强度均弱化集合种群的稳定性; 噪声关联强度强化集合种群的稳定性. 两噪声之间负关联时, 平均灭绝时间是加性噪声强度和乘性噪声强度的减函数, 是噪声关联时间的增函数; 两噪声之间正关联时, 平均灭绝时间是加性噪声强度和噪声关联时间乘性噪声强度的减函数, 是乘性噪声强度的非单调函数. 关键词： 集合种群 色交叉关联噪声 稳定性 平均灭绝时间     

Stochastic resonance in a bistable system with coloured correlation between additive and multiplicative noise

The phenomenon of stochastic resonance (SR) in a bistable nonlinear system with coupling between additive and multiplicative noises is investigated when the correlation between two noise terms is coloured. It is found that the signal-to-noise ratio (SNR) of the system is affected not only by the coupling strength λ between two noise terms, but also by the noise correlation time τ. The SNR is changed from a single peak, to two peaks with a dip, and then to a monotonically decreasing function with noise strength. The dependence of the SR on the initial conditions is entirely caused by the coupling strength λ between two noise terms.     

Stochastic resonance in linear system driven by multiplicative and additive noise

The stochastic resonance (SR) is studied in an overdamped linear system driven by multiplicative and additive noise when the additive noise is a linear combination of an asymmetric dichotomous noise and its square. The exact expressions are obtained for the first two moments and the correlation function and the SR phenomenon appeared. There are three different forms of SR: the bona fide SR, the conventional SR and SR in the broad sense. Moreover, the asymmetry of multiplicative noise has different effect on signal-to-ratio (SNR) for the first two different forms of SR and the effects of multiplicative noise and additive noise on SNR are different.     

Stochastic resonance in linear system driven by multiplicative noise and additive quadratic noise

In this paper the stochastic resonance (SR) is studied in an overdamped linear system driven by multiplicative noise and additive quadratic noise. The exact expressions are obtained for the first two moments and the correlation function by using linear response and the properties of the dichotomous noise. SR phenomenon exhibits in the linear system. There are three different forms of SR: the bona fide SR, the conventional SR and SR in the broad sense. Moreover, the effect of the asymmetry of the multiplicative noise on the signal-to-noise ratio (SNR) is different from that of the additive noise and the effect of multiplicative noise and additive noise on SNR is different.     

Effects of time delay on stochastic resonance of a periodically driven linear system with multiplicative and periodically modulated additive white noises

Stochastic resonance (SR) of a periodically driven time-delayed linear system with multiplicative white noise and periodically modulated additive white noise is investigated. In the condition of small delay time, an approximate analytical expression of output signal-to-noise ratio (SNR) is obtained. The analytical results indicate that (1) there exists a resonance peak in the curve for SNR versus time delay; (2) the time delay will suspend the SR dramatically for SNR versus other parameters of the system, such as noise intensity, correlation intensity, and signal frequency, once a certain value is reached, the SR phenomenon disappears.     

Impact of Colored Noise on Population Model with Allee Effect

We study a population model with strong and weak Allee effect driven by internal noise and external noise.Firstly, a single-species population model with Allee effect under environmental colored noise is established, then stable and unstable states are analyzed and interpreted in biology. After that, stationary probability distribution(SPD) of population is derived based on Fokker-Planck equation. Next, mean first-passage time(MFPT) is defined in order to quantify the transition between extinction state and survival state with Allee effect. It is found that population will not extinct when weak Allee effect exists. It is not beneficial to survival of the population with the increase of Allee threshold no matter whether strong Allee effect or weak Allee effect. When strong Allee effect occurs, the correlation time of multiplicative noise plays a positive role in survival of population, while the correlation time of additive noise has a negative effect. Crucially, the phenomenon of resonant activation is firstly discovered in population dynamics with Allee effect. The conclusions we obtain can be applied to the further research of population dynamics in ecology.