Stability and Bifurcation Analysis of Cyclic Genetic Regulatory Networks with Mixed Time Delays

Abstract

This paper analyzes the stability and bifurcation criteria of cyclic genetic regulatory networks with mixed time delays (discrete and Gamma-type distributed delays). It is of more realistic biological background to use Gamma-type distributed delay kernels to describe the delays in the process between genes. The key aim of our research is to reveal more about the dynamic functions played by biochemical parameters and different mechanisms of discrete and distributed time delays. The existence of positive equilibria in this kind of genetic network is verified. With the mathematical tools of subharmonic function and complex theory, exact conditions of biochemical parameters for stability and bifurcations in cyclic genetic regulatory networks with both positive and negative gains are deduced, respectively. Finally, several simulation examples are adopted to deepen our analysis.

State bounding and controller design for genetic regulatory networks with multiple delays and bounded disturbances

“…Let (c1) (7) and ( 8) hold for some scalar 𝛾 ∈ R > and vectors p, q ∈ R n > . (c2) (27) holds for some scalar 𝛾 ∈ R > .…”

“…In order to be consistent with practice, time delays include not only discrete ones but also distributed ones 1 . The stability analysis of GRNs with discrete delays and distributed delays greatly stimulate the interest of researchers, and a great deal of very important results have been achieved (see References 2‐13 and the references therein). On the other hand, environment fluctuations may influence the dynamics of GRNs, which also need to be taken into account in the analysis and design of GRNs.…”

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

“…However, so far, there are few results on the study of bifurcation for gene expression processes which have distributed delays. Distributed delays in Gamma-type were incorporated in a cyclic gene expression network [35], and the bifurcation and oscillation were discussed. But their model does not take sRNAs into account, which is depicted in the second equation in system (2).…”

Dynamics analysis for a discrete dynamic competition model

“…Recently, many research papers suggested that the mathematical model of economic system dynamics is more realistic and appropriate when it is modeled by discrete-time equations. The dynamics of the discrete-time models can exhibit much richer dynamics than those observed in continuous-time counterparts and can lead to chaotic behaviors [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”