  Numerous studies have demonstrated the important role of noise in the
dynamical behaviour of a complex system. The most probable trajectories of
nonlinear systems under the influence of Gaussian noise have recently been
studied already. However, there has been only a few works that examine how most
probable trajectories in the two-dimensional system (MeKS network) are
influenced under non-Gaussian stable L\'evy noise. Therefore, we discuss the
most probable trajectories of a two-dimensional model depicting the competence
behaviour in B. subtilis under the influence of stable L\'evy noise. On the
basis of the Fokker-Planck equation, we describe the noise-induced most
probable trajectories of the MeKS network from the low ComK protein
concentration (vegetative state) to the high ComK protein concentration
(competence state) under stable L\'evy noise. We demonstrate choices of the
non-Gaussianity index $\alpha$ and the noise intensity $\epsilon$ which
generate the ComK protein escape from the low concentration to the high
concentration. We also reveal the optimal combination of both parameters
$\alpha$ and $\epsilon$ making the tipping time shortest. Moreover, we find
that different initial concentrations around the low ComK protein concentration
evolve to a metastable state, and provide the optimal $\alpha$ and $\epsilon$
such that the distance between the deterministic competence state and the
metastable state is smallest.

Dynamical Systems;Molecular Networks