| Genetic regulatory networks(GRNs) are the interaction networks of DNAs、RNAs、proteins and metabolites in cells which are in the regulation process of genes,and they have been one of the important research areas of biological mathematics.In GRNs, Markov chain is widely applied in the model as a kind of transformation mechanism. The biggest advantage of Markov jumping parameters is that they can simulate the changes of system construction, for example, the damage or repairation to components, the sudden change of environments, the change of subsystem connection each other, and so on. Therefore, the research of systems with Markov jumping parameters has been a hot topic. The main contents of the paper are as follows:The first, in allusion to GRNs with time-varying delays and Markovian jumping parameters, we research the asymptotic stability problem of the kinds of systems. By constructing appropriate Lyapunov–Krasovskii functional, sufficient conditions for asymptotic stability of the systems are given in the form of linear matrix inequality(LMI). And then, numerical examples have been presented that the method is better than one given in[Math. Probl. in Engineer., 2012, 2012(4): 351–361].The second, in allusion to GRNs with delays in both the continuous-time case and the discrete-time case, the global exponential stability problems are investigated.First, for continuous-time case, by constructing an appropriate Lyapunov–Krasovskii functional and using the Dini derivative method the global exponential stability criteria are obtained. Second, by using the semi-discretization method and the IMEX- θ method, two discrete-time GRN models are derived. It is shown that under the same sufficient prerequisites, these two discrete-time GRN models are globally exponentially stable. Compared with [Appl. Math. Comput, 2013, 220(2):507–517], it is proved that our method is so much better. Finally, a pair of numerical examples are given to show the validity of the obtained theoretical results. |
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