Stability of Gene Regulatory Networks Modeled by Generalized Proportional Caputo Fractional Differential Equations

Ricardo Almeida; Ravi P Agarwal; Snezhana Hristova; Donal O'Regan

doi:10.3390/e24030372

Stability of Gene Regulatory Networks Modeled by Generalized Proportional Caputo Fractional Differential Equations

Entropy (Basel). 2022 Mar 5;24(3):372. doi: 10.3390/e24030372.

Authors

Ricardo Almeida¹, Ravi P Agarwal², Snezhana Hristova³, Donal O'Regan⁴

Affiliations

¹ Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal.
² Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA.
³ Faculty of Mathematics and Informatics, Plovdiv University "P. Hilendarski", 4000 Plovdiv, Bulgaria.
⁴ School of Mathematical and Statistical Sciences, National University of Ireland, H91 TK33 Galway, Ireland.

Abstract

A model of gene regulatory networks with generalized proportional Caputo fractional derivatives is set up, and stability properties are studied. Initially, some properties of absolute value Lyapunov functions and quadratic Lyapunov functions are discussed, and also, their application to fractional order systems and the advantage of quadratic functions are pointed out. The equilibrium of the generalized proportional Caputo fractional model and its generalized exponential stability are defined, and sufficient conditions for the generalized exponential stability and asymptotic stability of the equilibrium are obtained. As a special case, the stability of the equilibrium of the Caputo fractional model is discussed. Several examples are provided to illustrate our theoretical results and the influence of the type of fractional derivative on the stability behavior of the equilibrium.

Keywords: Lyapunov functions; equilibrium; generalized exponential stability; generalized proportional Caputo fractional derivatives; model of gene regulatory networks.