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

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.

Highlights

Gene expression is the process where the hereditary code of a gene is used for synthesizing proteins and producing the structures of the cell
From Lemma 1, it follows that the generalized proportional Caputo fractional derivative of a nonzero constant is not zero, and applying Corollary 1, we introduce the following definition
The goal of our paper is to study the exponential and asymptotic stability of the equilibrium of (9); equivalently, we study the stability properties of the zero solution of the IVP for FrDE (11)

Summary

Introduction

Gene expression is the process where the hereditary code of a gene is used for synthesizing proteins and producing the structures of the cell. Mathematical models of gene regulatory networks are described and studied in several papers (see, for example, [1,2], for fractional order [3–6], and with delays [7,8]). A gene regulated model with the generalized proportional Caputo fractional derivative is set up, and the equilibrium is defined. The generalized exponential stability is introduced and studied via the application of Lyapunov functions and their generalized Caputo proportional fractional derivatives. Generalized proportional Caputo fractional derivatives were recently introduced (see [12,13]); this type of derivative is a generalization of the Caputo fractional derivative, and their application provides wider possibilities for modeling adequately the complexity of real-world problems. Some properties of absolute values of Lyapunov functions and their fractional derivatives are discussed, and several examples are provided to illustrate the properties. Several examples are provided to illustrate the theoretical results and the dependence of the fractional derivative on the behavior of the solutions

Notes on Fractional Calculus

Statement of the Problem

Applications

Conclusions