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Dynamical behavior of a stochastic model of gene expression with distributed delay and degenerate diffusion |
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| Authors: | Qun Liu Tasawar Hayat Ahmed Alsaedi |
| Affiliation: | 1. School of Mathematics and Statistics, Key Laboratory of Applied Statistics of MOE, Northeast Normal University, Changchun, Jilin Province, PR China;2. School of Mathematics and Statistics, Guangxi Colleges and Universities Key Laboratory of Complex System, Optimization and Big Data Processing, Yulin Normal University, Yulin, Guangxi, PR China;3. Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, King Abdulaziz University, Jeddah, Saudi Arabia;4. Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan |
| Abstract: | This article is concerned with a stochastic model of gene expression with distributed delay and degenerate diffusion. We transform the model with weak kernel case into an equivalent system through the linear chain technique. Since the diffusion matrix is of degenerate type, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We prove the densities of the distributions of the solutions can converge in L1 to an invariant density. The existence of the stationary distribution implies stochastic weak stability. Numerical simulation is introduced to illustrate the analytical result. |
| Keywords: | Gene expression stochastic perturbations Markov semigroups distributed delay stationary distribution |