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Gene transcriptional regulatory is an inherently noisy process. In thispaper, the study of fluctuations in a gene transcriptional regulatory system isextended to the case of L\'evy noise, a kind of non-Gaussian noises which candescribe unpredictable jump changes of the random environment. The stationaryprobability density is given to explore the key roles of L\'evy noise in thegene regulatory networks. The results demonstrate that the parameters of L\'evynoise, including noise intensity, stability index and skewness parameter caninduce switches between distinct gene-expression states. A further concern isthe switching time (from the high concentration state to the low concentrationone or from the low concentration state to the high concentration one), whichis a random variable and often referred to as the mean first passage time. Theeffects of L\'evy noise in expression time and degradation time is studied bycomputing the mean first passage time in two directions and a number ofdifferent peculiarities of non-Gaussian L\'evy noise compared with Gaussiannoise are observed.

L\'evy Noise Induced Switch in the Gene Transcriptional Regulatory System