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