Abstract
This paper investigates a class of fractional-order delayed impulsive gene regulatory networks (GRNs). The proposed model is an extension of some existing integer-order GRNs using fractional derivatives of Caputo type. The existence and uniqueness of an almost periodic state of the model are investigated and new criteria are established by the Lyapunov functions approach. The effects of time-varying delays and impulsive perturbations at fixed times on the almost periodicity are considered. In addition, sufficient conditions for the global Mittag–Leffler stability of the almost periodic solutions are proposed. To justify our findings a numerical example is also presented.
Highlights
The regulation of the genes’ expression in the process of operating of organisms on the molecular level is mainly realized via genetic regulatory systems organised as networks of connections between DNA, RNA, proteins, and small molecules
Ordinary differential equations are among the proposed formalisms, that are intensively applied by numerous researchers in modelling of gene regulatory networks (GRNs) [2,3,4]
Investigating the effects of time delays is very important in the models of GRNs in order to understand the transcriptional process of genetic gene and analyze the limited speed of gene transcription
Summary
The regulation of the genes’ expression in the process of operating of organisms on the molecular level is mainly realized via genetic regulatory systems organised as networks of connections between DNA, RNA, proteins, and small molecules. An impulsive control strategy is considered for a class of fractional-order GRNs with time-varying delays; The almost periodicity notion is introduced to the model under consideration which initiates the development of the almost periodicity theory for impulsive fractional. GRNs; New existence and uniqueness results for the almost periodic states are established; New criteria for global Mittag–Leffler stability of an almost periodic state of the impulsive model under consideration are proved; We apply an extended Lyapunov function approach which allows representing the obtained results in terms of the model’s parameters, and leads to a better exploration of the impulsive effect. We will define the impulsive GRN model of fractional order with time-varying delays as follows:. The model (2) is an extension of the GRN models proposed in [5,6] and some others, considering different time-varying delays, fractional-order derivatives, and impulsive effects.
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