Stability Of Gene Regulatory Networks With Lévy Noise

In recent years, with the development of biomedical science and bioinformatics,the research on the performance of gene regulatory network is highly concerned by scientists. The protein, DNA, RNA and other molecules in the gene regulation network play an important role in the regulation of various mechanisms of biological systems, so the analysis and relative comprehensive conclusions of the changes of molecular concentration is undoubtedly the hot focus. At present, as respect to theoretical model, the gene regulatory networks described by differential equations are widely used and mainly involved in the stochastic hybrid gene regulatory networks and the gene regulatory networks actuated by the Wiener process. It has been proved by experiments that some molecular concentration changes under the influences of some specific factors, when the molecular concentration changes continuously with time, mutation will occur with the concentration of some molecular. That is to say, at this time the gene regulatory network with Wiener process will make the expression of the networks less accurate. In order to analyze the effect of the molecular concentration mutations on the properties of gene regulatory networks accurately, we intend to use the gene regulatory network actuated by Lévy noise for analyzing whether the changes of relevant network performance are within controllable range when mutations of molecular concentration take place.1. We investigated the stability of the gene regulatory networks with Lévy noise and time-varying delays. By constructing a Lyapunov function, using the generalized Ito formula, the Leibniz-Newton formula, Kunita estimation and basic inequalities for the estimation of mutated items, we obtain sufficient conditions for the global asymptotic stability of gene regulatory network with Lévy noise and time-varying delays, and then prove the effectiveness of the sufficient condition by simulations.2. We studied the boundness of the gene regulatory networks with Lévy noise and Markov modulation. Taking the accidental phenomenon into consideration that molecular concentration explosion of the gene regulatory networks with Markov modulation may occur, we get the supremum with the help of appropriate Lyapunov functions, generalized Ito formula, some complex skills and by making full use of the Lévy noise to suppress possible molecular concentration explosive phenomenon in turn, and finally by applying the numerical simulation results to verify the feasibility of the conclusion.3. We considered the exponential stability of the gene regulatory networks with Lévy noise and distributed parameters. Considering describe the uneven distribution of molecular concentration by the distributed parameters, we get the sufficient exponential stability conditions of the gene regulatory networks stability with Lévy noise and distributed parameters by constructing a Lyapunov function on the space variable average, using the Hardy-Poincare inequality and the design of the feedback controller in order to ensure that the conclusions related to coefficient of diffusion,and the correctness of the conclusion was verified by numerical simulations.