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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 τ and τ 0 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 α and the correlation time τ of the multiplicative noise will always magnify the SR phenomenon. It is worth to mention that the correlation time τ 0 of the additive noise can stimulate the SR effect, but not alter the maximum of the SNR. [ABSTRACT FROM AUTHOR]

In this paper, the regime shift behaviors between the prosperous state and the extinction state and stochastic resonance (SR) phenomenon for a metapopulation system subjected to time delay and correlated Gaussian colored noises are investigated. Through the numerical calculation of the modified potential function and the stationary probability density function (SPDF), one can make clearly the following results: Both multiplicative noise and noise correlation times can improve effectively the ecological stability and prolong the survival time of the system; while additive noise, time delay and noise correlation strength can weaken significantly the biological stability and speed up the extinction of the population. As for the signal-to-noise ratio (SNR), it is found that time delay, multiplicative noise and noise correlation strength can all impair the SR effect. Conversely, the two noise correlation times and additive noise are in favor of the improvement of the peak values of SNR. It is particularly worth mentioning that in the case of M = 0. 0 3 , time delay α and self-correlation time τ 2 of the additive noise display exactly the opposite effect on the stimulation of the resonant peak in the SNR– λ plots. [ABSTRACT FROM AUTHOR]

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 τ can increase the probability density around both stable states and consolidate the stability of the neural system. Furthermore, the other noise correlation time τ 0 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 α and the departure parameter q will always amplify the SR phenomenon. Moreover, the noise correlation time τ 0 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 τ 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 τ. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract.: In this paper, the modified potential function, the stationary probability distribution function (SPDF), the mean growth time and the mean degeneration time for a vegetation growth system with time delay are investigated, where the vegetation system is assumed to be disturbed by cross-correlated multiplicative and additive noises. The results reveal some fact that the multiplicative and additive noises can both reduce the stability and speed up the decline of the vegetation system, while the strength of the noise correlation and time delay can both enhance the stability of the vegetation and slow down the depression process of the ecological system. On the other hand, with regard to the impacts of noises and time delay on the mean development and degeneration processes of the ecological system, it is discovered that 1) in the development process of the vegetation population, the increase of the noise correlation strength and time delay will restrain the regime shift from the barren state to the boom one, while the increase of the additive noise can lead to the fast regime shift from the barren state to the boom one. 2) Conversely, in the depression process of the ecological system, the increase of the strength of the correlation noise and time delay will prevent the regime shift from the boom state to the barren one. Comparatively, the increase of the additive and multiplicative noises can accelerate the regime shift from the boom state to the barren state.Graphical abstract: [ABSTRACT FROM AUTHOR]

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. • A stochastic metapopulation system is established. • The modified potential function and the SPDF is analyzed. • The SR phenomenon for the population system is discussed. [ABSTRACT FROM AUTHOR]