| Stochastic fluctuation phenomenon is widespread and pervasive in our familiar macroscopic world. Fluctuations of various physical quantities in nature are usually described by noises. Noises can be divided into internal and external noise depending on the origin of the noise. The internal noise is caused by the internal dynamics of the system. Revealing the important effects of random forces generated by nonlinear conditions, has become an important part of the current development of statistical physics and nonlinear science. In this paper, using the Langiven equation, we have studied some statistical characteristics of the bitable syetem and metapopulation system. And the main work is divided into the following sections.1. By virtue of the Liouville Equation and Novikov Theorem, the approximate Fokker-Planck equation is derived, and the analytic expression of the stationary probability distribution is obtained in an asymmetric bistable system driven by cross-correlated multiplicative white noise and additive white noise with periodic signal. Based on the computed results, the effects of the following parameters, including the intensity of noises and their correlation, the amplitude and frequency of the periodic signal, and the asymmetric parameter of the system on the stationary probability distribution are investigated. By defining the function R=D/Q, effects of the amplitude and frequency of the periodic signal on the SPD and moments are studied in three cases (R=1, R>1, R<1).2. In this paper, the stability for a metapopulation system subjected to colored cross-correlated colored-noises is investigated based on the Levins model. By virtue of a unified colored-noise approximation approach, the approximate Fokker-Planck equation is obtained, and the steady-state probability density function is obtained by solving the Fokker-Planck equation. And then, the analytical expression of the mean first-passage time of the system is derived by using the steepest descent method. The result shows:|λ|strengthen the stability of the system in the case of λ>0;for the case of λ<0,|λ|weaken the stability of the system; the decreasing of Q strengthen the stabiliy of the system when the value of Q is less than0.6; the mean extinction time is a increasing function of D(Ï„), but a decreasing function of Q.3. In this paper, we finished the following work based on the Levins model.Firstly, the stability and mean extinction time for a metapopulation system driven by cross-correlated noises is investigated based on the Levins model with habitat destruction. The stationary probability distribution is derived according to the Fokker-Planck equation. By virtue of the steepest decent method, the explicit expression of the mean extinction time is obtained. The result shows that:Q and D weaken the stabiliy of the system, but λ strengthen the stabiliy of the system; The mean extinction time T(xsâ†'xo) is a non-monotonic function of the multiplicative noise intensity, and there exists a maximal value as the multiplicative noise intensity varies. Secondly, the stability and mean extinction time for a metapopulation system driven by cross-correlated noises are investigated based on the Levins model. The stationary probability distribution is derived according to the Fokker-Planck equation. By virtue of the steepest decent method, the explicit expression of the mean extinction time is obtained. The numerical results show that Q and D weaken the stability of metapopulation. On the contrary, the stability of metapopulation is strengthed as Ï„ increasing. Furthermore, the mean extinction time T(xsâ†'xo) is a decreasing function of D when λ<0, while a non-monotonic function of D when λ>0, and there exists a maximal value as D varies. Meantime, T(Xsâ†'xo) is a increasing function of r.Thirdly, the stability and mean extinction time for a metapopulation system driven by cross-correlated colored noises are investigated based on the classical Levins model with habitat destruction. The stationary probability distribution is derived according to the Fokker-Planck equation. By virtue of the steepest decent method, the explicit expression of the mean extinction time is obtained. The numerical results show that: D strengthen the stability of metapopulation; Q weaken the stability of metapopulation, the stability of metapopulation is strengthened as Ï„1increases; with the increasing of Ï„2, the stability of metapopulation is strengthened; the stability of metapopulation is strengthened as A increasing; the mean extinction time T(xsâ†'xo) is a increasing function of Ï„1; T(xsâ†'xo) is a increasing function of r2.4. By virtue of Liouville Theorem and unified colored-noise approximation approach, an approximate Fokker-Planck equation for a tree growth Logistic model subjected to cross-correlated colored noises is derived, and the steady-state probability distribution function is obtained. The steady-state properties of the Logistic model are analyzed. We find the following:The position of peak of SPD moves toward left side as D increases while the position of the peak moves toward the contrary direction with Q increasing. The peak of SPD becomes narrow and grows in height as|λ|increases, and for the case of λ>0,the position of peak moves toward right as D increases, but it is opposite for the case of λ<0as Q increases. In the above discussed problems, our result is valuable for understanding the impact of external environment on the tree growth if we considering the noise as the flucation of the environment. |
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