Gene transcription and protein synthesis can be widely found in
biological research. Based on this viewpoint, this article focuses on a
two-dimensional stochastic delay differential equation (SDDE) to
investigate the gene expression. A wealth of dynamic behaviors from a
mathematical perspective are observed. An optimized method to handle the
stochastic delayed system is proposed and we applied it to the model in
this paper. Then an averaging Ito equation is obtained. Based on this,
we studied the stochastic bifurcation of the system and the
corresponding probability distribution with time delay, degradation
rate, the intensity of the stochastic process, and the randomization
degree as parameters. The numerical results revealed the transition of
the probability density function of the system from a Dirac δ function
to a nontrivial stationary state. In addition, stochastic P-bifurcation
occurs when the presentation of the stationary probability density
change from singular peak to crater.
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