Stability and bifurcation analysis of a gene expression model with small RNAs and mixed delays

Fan Qing,Zhengxin Wang,Chengdai Huang,Jianlong Qiu,Jinxing Lin,Guoping Jiang,Min Xiao,Cong Zheng

doi:10.1186/s13662-019-2180-7

Abstract

This paper investigates a gene expression model, which is mediated by sRNAs (small RNAs) and includes discrete and distributed delays. We take both the strong and weak kernel forms of distributed delay into consideration. The discrete time delay is chosen as the bifurcation parameter. By analyzing the distribution of characteristic values, we obtain the sufficient conditions of stability and examine the existence of periodic oscillations. When the discrete time delay is small and not greater than the threshold, the equilibrium of the gene expression model is asymptotically stable. When the bifurcation parameter exceeds the critical value, the model can produce limit cycles. Finally, numerical simulations are implemented to verify the correctness of our theoretical results.

Highlights

The study of bifurcation phenomena has aroused the interest of scientists in many fields
In order to describe the genetic process of organisms more accurately, we introduce the distributed delays to exactly describe the change of the time delay in the reality
The introduction of distributed time delay increases the dimension of the network and makes the dynamic behaviors more complex

Summary

Introduction

The study of bifurcation phenomena has aroused the interest of scientists in many fields. Distributed delays in Gamma-type were incorporated in a cyclic gene expression network [35], and the bifurcation and oscillation were discussed Their model does not take sRNAs into account, which is depicted in the second equation in system (2). The influence of distributed time delays on dynamical behaviors of a mathematical model of gene expression was studied [36]. Both the cases of the weak and strong delay kernels were addressed. In the case of weak kernel, the stability and local bifurcation of the gene expression model with sRNAs and mixed delays are discussed in Sect.

Then we have

We denote

It follows that

Conclusion