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Important effects of noise on a one-dimensional gene expression modelinvolving a single gene have recently been discussed. However, few works havebeen devoted to the transition in two-dimensional models which include theinteraction of genes. Therefore, we investigate here, a quantitativetwo-dimensional model (MeKS network) of gene expression dynamics describing thecompetence development in the B. subtilis under the influence of L\'evy as wellas Brownian motions, where noises can do the B. subtilis a favor in nutrientdepletion. To analyze the transitions between the vegetative and the competenceregions therein, two deterministic quantities, the mean first exit time (MFET)and the first escape probability (FEP) from a microscopic perspective, as wellas their averaged versions from a macroscopic perspective, are applied. Therelative contribution factor (RCF), the ratio of non-Gaussian and Gaussiannoise strengths, is adopted to implement optimal control in these transitions.Schematic representations indicate that there exists an optimum choice thatmakes the transition occurring at the highest probability. Additionally, we usea geometric concept, the stochastic basin of attraction, to exhibit a pictorialcomprehension about the influence of the L\'{e}vy motion on the basin stabilityof the competence state.

L\'{e}vy noise-induced transitions in gene regulatory networks