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The multi-stable dynamics behavior for molecular models of mRNA and proteins in regulatory mechanisms has received increasing attention based on these biological findings. In this work, we focus on the global Mittag–Leffler boundedness (GMLB) for discrete-time fractional-order fuzzy delayed complex-valued genetic regulatory networks (DTFFDCVGRNs). To guarantee GMLB of the addressed molecular models, various sufficient criteria, including algebraic conditions and the condition in terms of LMI, are derived based on Caputo fractional difference operator using the Lyapunov function approach and the discrete Laplace transform. In addition, estimates of various global Mittag–Leffler attraction sets (GMLASs) known as convergence balls are provided. Finally, a numerical example demonstrates the proposed theoretical results. • Using IF-THEN rules, this study attempts to solve the GMLB problem of DTFFDCVGRNs for the first time. • We derive some effective criteria based on Lyapunov function theory and discrete-time fractional calculus theory for real-valued algebraic inequalities and GMLASs to realize the GMLB of DTFFDCVGRNs. • Based on discrete-time fractional inequality theory C ∇ 0 δ (θ (ħ) ¯ θ (ħ)) ≤ C ∇ 0 δ θ (ħ) ¯ θ (ħ) + θ (ħ) ¯ C ∇ 0 δ θ (ħ) , δ ∈ (0 , 1) , an important part of the complex domain, which improves the given results in You et al. (2020) is constructed. • Different from the existing works such as fractional-order GRNs (Huang et al., 2016, Zhang et al., 2019, Chen et al., 2023, Qiao et al., 2020, Stamova and Stamova, 2021, Ren et al., 2015) and fractional-order CV-GRNs with a real decomposition approach (Wu et al., 2020), this paper introduces DTFFDCVGRNs via direct complex field approaches, which have the lower dimension and complex-valued LMI than some other decomposition approaches. [ABSTRACT FROM AUTHOR]

The synchronization problem for a nonlinear dynamic fuzzy modeling of fractional-order genetic regulatory networks with adaptive feedback controllers is addressed in this article. To address such a problem, a fuzzy feedback MAX template and a fuzzy feedback MIN template of fuzzy modeling are introduced. We developed adaptive based criteria to ensure the asymptotic and finite-time stability of the error system by constructing a suitable Lyapunov functional with linear matrix inequality (LMI) approach, which ensures the drive system synchronizes with the response system. Meanwhile, two distinct controllers, linear feedback and adaptive feedback control are being developed. It should be noted that the presented results are expressed in terms of LMI, which can be efficiently solved using Matlab LMI toolbox. Numerical simulations are carried out to illustrate the theoretical results. • The adaptive synchronization of nonlinear dynamic FOFGRNs with time-varying delay is investigated. • Sufficient conditions are established to ensure that the FOFGRNs achieve asymptotic and finite-time. synchronization. • To account for ambiguity, the fuzzy theory is incorporated into the FOGRNs. • Based on fractional Lyapunov stability theory and the LMI technique, the asymptotic and finite-time criteria are derived. • The obtained LMI stability criteria have a simpler form than the algebraic stability criteria proposed in the literature. [ABSTRACT FROM AUTHOR]