Stability and Hopf bifurcation in a model of gene expression with distributed time delays

We discuss the existence of Hopf bifurcation, the dynamical stability breaking of the molecular concentrations of two generic biochemical reaction systems and their sensitivity to the changes of the parameters incorporated in the model equations. It is found that the oscillatory dynamics can be expected for the systems due to the inclusion of time delays. The chaotic dynamics and the periodic windows in the chaotic domains can exist in the case of the system with two time delays. The proposed mathematical method may have the significance in the problems where the negative and/or positive feedback dynamics, as well as the time delays have the characteristic physical and biological background.

Download Full-text

Hopf bifurcation in continuous-time systems with distributed time delays

Frequency-Domain Approach to Hopf Bifurcation Analysis - World Scientific Series on Nonlinear Science Series A ◽

10.1142/9789811205477_0007 ◽

2019 ◽

pp. 269-307

Keyword(s):

Hopf Bifurcation ◽

Continuous Time ◽

Time Delays ◽

Continuous Time Systems ◽

Distributed Time Delays ◽

Time Systems

Download Full-text

Stability and Hopf Bifurcation Analysis of a Gene Expression Model with Diffusion and Time Delay

Abstract and Applied Analysis ◽

10.1155/2014/738682 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9 ◽

Cited By ~ 6

Author(s):

Yahong Peng ◽

Tonghua Zhang

Keyword(s):

Gene Expression ◽

Hopf Bifurcation ◽

Local Stability ◽

Time Delays ◽

Center Manifold ◽

Hopf Bifurcations ◽

Expression Model ◽

The Stability ◽

And Diffusion ◽

Gene Expression Model

We consider a model for gene expression with one or two time delays and diffusion. The local stability and delay-induced Hopf bifurcation are investigated. We also derive the formulas determining the direction and the stability of Hopf bifurcations by calculating the normal form on the center manifold.

Download Full-text

Turing–Hopf bifurcation and spatiotemporal patterns in a Gierer–Meinhardt system with gene expression delay

Nonlinear Analysis Modelling and Control ◽

10.15388/namc.2021.26.23054 ◽

2021 ◽

Vol 26 (3) ◽

pp. 461-481

Author(s):

Shuangrui Zhao ◽

Hongbin Wang ◽

Weihua Jiang

Keyword(s):

Gene Expression ◽

Hopf Bifurcation ◽

Pattern Formation ◽

Time Delays ◽

Spatiotemporal Patterns ◽

Third Order ◽

Spatially Inhomogeneous ◽

Math Biol ◽

Formation Systems ◽

Expression Time

In this paper, we consider the dynamics of delayed Gierer–Meinhardt system, which is used as a classic example to explain the mechanism of pattern formation. The conditions for the occurrence of Turing, Hopf and Turing–Hopf bifurcation are established by analyzing the characteristic equation. For Turing–Hopf bifurcation, we derive the truncated third-order normal form based on the work of Jiang et al. [11], which is topologically equivalent to the original equation, and theoretically reveal system exhibits abundant spatial, temporal and spatiotemporal patterns, such as semistable spatially inhomogeneous periodic solutions, as well as tristable patterns of a pair of spatially inhomogeneous steady states and a spatially homogeneous periodic solution coexisting. Especially, we theoretically explain the phenomenon that time delay inhibits the formation of heterogeneous steady patterns, found by S. Lee, E. Gaffney and N. Monk [The influence of gene expression time delays on Gierer–Meinhardt pattern formation systems, Bull. Math. Biol., 72(8):2139–2160, 2010.]

Download Full-text

Oscillatory Dynamics of P53 Network With Time Delays

Combinatorial Chemistry & High Throughput Screening ◽

10.2174/1386207321666180601080026 ◽

2018 ◽

Vol 21 (6) ◽

pp. 411-419 ◽

Cited By ~ 2

Author(s):

Conghua Wang ◽

Fang Yan ◽

Yuan Zhang ◽

Haihong Liu ◽

Linghai Zhang

Keyword(s):

Hopf Bifurcation ◽

Cell Fate ◽

Time Delays ◽

Sustained Oscillation ◽

Multiple Time ◽

Cell Fate Decisions ◽

Oscillatory Dynamics ◽

Main Components ◽

Parameter Values ◽

P53 Network

Aims and Objective: A large number of experimental evidences report that the oscillatory dynamics of p53 would regulate the cell fate decisions. Moreover, multiple time delays are ubiquitous in gene expression which have been demonstrated to lead to important consequences on dynamics of genetic networks. Although delay-driven sustained oscillation in p53-based networks is commonplace, the precise roles of such delays during the processes are not completely known. Method: Herein, an integrated model with five basic components and two time delays for the network is developed. Using such time delays as the bifurcation parameter, the existence of Hopf bifurcation is given by analyzing the relevant characteristic equations. Moreover, the effects of such time delays are studied and the expression levels of the main components of the system are compared when taking different parameters and time delays. Result and Conclusion: The above theoretical results indicated that the transcriptional and translational delays can induce oscillation by undergoing a super-critical Hopf bifurcation. More interestingly, the length of these delays can control the amplitude and period of the oscillation. Furthermore, a certain range of model parameter values is essential for oscillation. Finally, we illustrated the main results in detail through numerical simulations.

Download Full-text

Optimal harvesting strategies of a stochastic competitive model with S-type distributed time delays and Lévy jumps

Boundary Value Problems ◽

10.1186/s13661-021-01509-6 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Hong Qiu ◽

Wenmin Deng ◽

Mingqi Xiang

Keyword(s):

Time Delays ◽

Sufficient Conditions ◽

Volterra Model ◽

Optimal Harvesting ◽

Ergodic Method ◽

Technical Assumption ◽

Distributed Time Delays ◽

Competitive Model ◽

The Stability ◽

Lévy Jumps

AbstractThe aim of this paper is to investigate the optimal harvesting strategies of a stochastic competitive Lotka–Volterra model with S-type distributed time delays and Lévy jumps by using ergodic method. Firstly, the sufficient conditions for extinction and stable in the time average of each species are established under some suitable assumptions. Secondly, under a technical assumption, the stability in distribution of this model is proved. Then the sufficient and necessary criteria for the existence of optimal harvesting policy are established under the condition that all species are persistent. Moreover, the explicit expression of the optimal harvesting effort and the maximum of sustainable yield are given.

Download Full-text