Estimating the nonlinear effects of an ecological system driven by Ornstein-Uhlenbeck noise

•An effective FPE for a dynamical system driven by multiple O-U noises is derived.•A Levins model driven by O-U noise and cross-correlated noise is constructed.•Noise enhanced stability phenomena is observed. In this paper, a simple ecological model driven by Ornstein-Uhlenbeck noise and cross-correlated noise is constructed based on the classical Levins model. Initially, an approximate Fokker-Planck Equation for a general dynamical system driven by multiple sources of Ornstein-Uhlenbeck and cross-correlated noise is obtained by using the unified colored-noise approximation method and the stochastic equivalent rules. Then, the effects of Ornstein-Uhlenbeck noise on the dynamical characteristics of the modified Levins model are investigated. The theoretical dynamical quantities, including the steady-state probability distribution function and the mean extinction time, respectively, are analyzed. It is found that the external noises can lead to a decrease in the probability of an occupied patch, while, the internal noise, the cross-correlated noise and the self-correlation time of the external and internal noises can cause an increase in the probability of an occupied patch. Further, the impact of the noise intensities on the mean extinction time is analyzed. It is especially worthy to note that as the intensities of the internal noise and the cross-correlated noise increase, a noise-enhanced stability phenomenon is induced. Moreover, the mean extinction time increases as the self-correlated time and the intensity of the cross-correlated noise increase, yet the mean extinction time decreases as the intensity of the internal noise increases. The validity of theoretical method is checked by numerical simulation.