Fast Descent Method Research Articles

Mean extinction time and stability for a metapopulation system subjected to correlated Gaussian and non-Gaussian noises

In this paper, we investigate the shift between the stable state of a large density and the extinction state, the mean extinction time for a metapopulation system induced by the associated Gaussian additive noise and non-Gaussian multiplicative noise. By applying the approximate Fokker–Planck equation, the fast descent method and the path integral approach, we obtain the expressions of the effective potential function and the mean extinction time. The numerical results show that the additive noise can damage the stability of the system, while the associated noise and the noise correlation times τ0 and τ can strengthen the stability of the population system. In the meantime, the maximum of the mean first-passage time appears in the meta-population system due to the impacts of different types of noises and their correlation time terms. It can effectively extend the extinction time of the metapopulation to enhance the strength of the correlation noise and two terms of noise correlation time. Conversely, the additive noise plays a negative role in preventing the extinction of the population. On the other side, a proper small multiplicative noise intensity can produce the positive effect on lengthening the life of the metapopulation.

Stochastic dynamical characteristics for a time-delayed insect outbreak model driven by correlated multiplicative and additive noises

In this paper, we numerically investigate the stability and transient properties of an insect outbreak model driven by correlated multiplicative, additive Gaussian noises and time delay. By using the Fokker–Planck equation, the small time delay method and the fast descent method, we analyze the mean reproduction and decline time of the insect growth system. Numerical results show that the resonant phenomenon of maximization of the mean first-passage time (MFPT) appears due to the joint action of all kinds of noises and time delay. It can hold down the insect population transfer from the stable state of a small number to the large one, enhance effectively the stability of the system and further inhibit the expansion of the insect population to increase noises intensities and time delay. In the process of restraining the expansion of the insect population, the increase of the association noise strength caused by the intrinsic noise and the extrinsic noise, and time delay make a beneficial contribution to restraining the outbreak of the insect population. Analogously, the appropriate multiplicative and additive noise intensities can play a positive role in the inhibition of pests. On the contrary, during the process of destroying a large number of insects, the increase of all noises and time delay can produce positive effects on controlling a plague of insects.

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.

Stability and mean extinction time for a metapopulation system subjected to two kinds of time delays and associated multiplicative and additive noises

In this paper, the stability and the mean extinction time for a double time-delayed metapopulation system driven by associated multiplicative and additive noises are investigated. By using the Fokker–Planck equation, the fast descent method and the method of small time delay, we obtain the expressions of stationary probability distribution and the mean extinction time. The numerical results demonstrate that the multiplicative, additive noises and time delay θ can destroy the stability of the system. Whereas, the associated noise and time delay τ can consolidate the population system. Meanwhile, the resonant phenomenon of the mean first-passage time (MFPT) occurs in the metapopulation system due to the joint action of different types of noises and time delays. The increase of the strength of associated noise and time delay τ can effectively prolong the extinction time of the metapopulation. Inversely, the additive noise plays a negative role in restraining the collapse of the population. On the other hand, the appropriate multiplicative noise intensity can also extend the life of the metapopulation system. However, the varying of time delay θ cannot change the maximum of the mean extinction time, but it could produce different effects on the MFPT under the conditions of different parameters.

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.

Stochastic Resonance for a Cancer Growth System Subjected to Correlated Multiplicative and Additive Noises

In the present paper, the phenomena of stochastic resonance (SR) for a stochastic cancer development system that is driven by correlated multiplicative and additive noises is investigated. By using the fast descent method and the two-state theory, the expression of the signal-to-noise ratio (SNR) is obtained. Numerical results show that conventional SR occurs in the cancer growth model under different values of the system parameters. If the correlation strength between the two noises is positive or negative, the effects of the addictive noise intensity, the multiplicative noise intensity, and the correlated noise strength on the SNR are different.

STOCHASTIC RESONANCE FOR A METAPOPULATION SYSTEM SUBJECTED TO CORRELATED COLORED NOISES

In the present paper, for a Levins metapopulation system that is driven by correlated colored noises, the phenomenon of stochastic resonance (SR) is investigated. Based on the two-state theory and by the use of fast descent method, the expression of the signal-to-noise ratio (SNR) is obtained. Via a numerical simulation, it is shown that the conventional SR occurs in the Levins model for the different values of system parameters. And furthermore, it is revealed that, under the different conditions that if the correlation intensities between the two noises are different, i.e. positive or negative, then all the effects of the addictive noise intensity, the multiplicative noise intensity, the correlated noise intensity and the correlation time on SNR are different.