Differential-Delay Equation Model of Gene Expression: A Continuum Approach

2008 ◽  
Author(s):  
Anael Verdugo ◽  
Richard H. Rand
Keyword(s):  
Analytical Study ◽  
Equation Model ◽  
Differential Delay Equation ◽  
Differential Delay ◽  
Differential Integral Equation ◽  
Gene Regulatory ◽  
The Stability ◽  
Delay Differential ◽  
Steady State Solutions ◽  
Nonlinear Delay

This paper presents an analytical study of the stability of the steady state solutions of a gene regulatory network with time delay. The system is modeled as a continuous network and takes the form of a nonlinear delay differential-integral equation coupled to an ordinary differential equation. Two examples are given in which the critical delay causing instability is computed.

DDE Model of Gene Expression: A Continuum Approach

2008 ◽  
Author(s):  
Anael Verdugo ◽  
Richard H. Rand
Keyword(s):  
Gene Expression ◽  
Regulatory Network ◽  
Analytical Study ◽  
Continuum Approach ◽  
Differential Integral Equation ◽  
Gene Regulatory ◽  
The Stability ◽  
Delay Differential ◽  
Steady State Solutions ◽  
Nonlinear Delay

This paper presents an analytical study of the stability of the steady state solutions of a gene regulatory network with time delay. The system is modeled as a continuous network and takes the form of a nonlinear delay differential-integral equation coupled to an ordinary differential equation. Two examples are given in which the critical delay causing instability is computed.

scholarly journals Non-Spiking Laser Controlled by a Delayed Feedback

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2069
Author(s):  
Anton V. Kovalev ◽  
Evgeny A. Viktorov ◽  
Thomas Erneux
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Optical Feedback ◽  
Laser Pulses ◽  
Equation Model ◽  
Rate Equations ◽  
Differential Equation Model ◽  
The Stability ◽  
Delay Differential ◽  
Intensity Laser

In 1965, Statz et al. (J. Appl. Phys. 30, 1510 (1965)) investigated theoretically and experimentally the conditions under which spiking in the laser output can be completely suppressed by using a delayed optical feedback. In order to explore its effects, they formulate a delay differential equation model within the framework of laser rate equations. From their numerical simulations, they concluded that the feedback is effective in controlling the intensity laser pulses provided the delay is short enough. Ten years later, Krivoshchekov et al. (Sov. J. Quant. Electron. 5394 (1975)) reconsidered the Statz et al. delay differential equation and analyzed the limit of small delays. The stability conditions for arbitrary delays, however, were not determined. In this paper, we revisit Statz et al.’s delay differential equation model by using modern mathematical tools. We determine an asymptotic approximation of both the domains of stable steady states as well as a sub-domain of purely exponential transients.

scholarly journals Effect of time delay on a detritus-based ecosystem

2006 ◽  
Vol 2006 ◽  
pp. 1-28 ◽  
Author(s):  
Nurul Huda Gazi ◽  
Malay Bandyopadhyay
Keyword(s):  
Time Delay ◽  
Equilibrium Points ◽  
Equation Model ◽  
Model System ◽  
Trophic Levels ◽  
Dynamical Analysis ◽  
Discrete Time Delay ◽  
Growth Equations ◽  
The Stability ◽  
Delay Differential

Models of detritus-based ecosystems with delay have received a great deal of attention for the last few decades. This paper deals with the dynamical analysis of a nonlinear model of a detritus-based ecosystem involving detritivores and predator of detritivores. We have obtained the criteria for local stability of various equilibrium points and persistence of the model system. Next, we have introduced discrete time delay due to recycling of dead organic matters and gestation of nutrients to the growth equations of various trophic levels. With delay differential equation model system we have studied the effect of time delay on the stability behaviour. Next, we have obtained an estimate for the length of time delay to preserve the stability of the model system. Finally, the existence of Hopf-bifurcating small amplitude periodic solutions is derived by considering time delay as a bifurcation parameter.

scholarly journals A Differential Equation Model of HIV Infection ofCD4+T-Cells with Delay

2008 ◽  
Vol 2008 ◽  
pp. 1-16 ◽  
Author(s):  
Junyuan Yang ◽  
Xiaoyan Wang ◽  
Fengqin Zhang
Keyword(s):  
T Cells ◽  
Time Delay ◽  
Hiv Infection ◽  
Equation Model ◽  
Feasible Region ◽  
Delay Model ◽  
Cure Rate ◽  
Differential Delay ◽  
Discrete Time Delay ◽  
The Stability

An epidemic model of HIV infection ofCD4+T-cells with cure rate and delay is studied. We include a baseline ODE version of the model, and a differential-delay model with a discrete time delay. The ODE model shows that the dynamics is completely determined by the basic reproduction numberR0<1. IfR0<1, the disease-free equilibrium is asymptotically stable and the disease dies out. IfR0>1, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region. In the DDE model, the delay stands for the incubation time. We prove the effect of that delay on the stability of the equilibria. We show that the introduction of a time delay in the virus-to-healthy cells transmission term can destabilize the system, and periodic solutions can arise through Hopf bifurcation.

OSCILLATIONS IN DELAY DIFFERENTIAL EQUATION MODEL OF REPRODUCTIVE HORMONES IN MEN WITH COMPUTER SIMULATIONS

1994 ◽  
Vol 02 (01) ◽  
pp. 73-90 ◽  
Author(s):  
PRITHA DAS ◽  
A.B. ROY
Keyword(s):  
Stable Limit Cycle ◽  
Equation Model ◽  
Reproductive Hormones ◽  
Phase Portraits ◽  
Stable Limit ◽  
Delay Model ◽  
Local Asymptotic Stability ◽  
The Stability ◽  
Linearized Model ◽  
Delay Differential

We produce here a delay model to explain the control of testosterone secretion. We have modified our earlier model by incorporating one negative feedback function which explains the inhibition of the pituitary secretion of the hormone LH (Luteinizing hormone) by the local testosterone concentration. We have derived the conditions for local asymptotic stability and switching to instability of the steady state. The length of the delay preserving the stability has also been derived. Lastly the conditions for instability and bifurcation results have been derived for the linearized model. Phase portraits of the original nonlinear model showing stable limit cycle have been simulated.

scholarly journals Stability of Stochastic Differential Delay Systems with Delayed Impulses

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Yanlei Wu
Keyword(s):  
Delay System ◽  
Delay Systems ◽  
Stochastic Delay ◽  
Differential Delay ◽  
Almost Sure Exponential Stability ◽  
Continuous Dynamic System ◽  
Continuous Dynamic ◽  
The Stability ◽  
Delay Differential ◽  
Delayed Impulses

We investigate the stability of stochastic delay differential systems with delayed impulses by Razumikhin methods. Some criteria on thepth moment and almost sure exponential stability are obtained. It is shown that an unstable stochastic delay system can be successfully stabilized by delayed impulses. Moreover, it is also shown that if a continuous dynamic system is stable, then, under some conditions, the delayed impulses do not destroy the stability of the systems. The effectiveness of the proposed results is illustrated by two examples.

scholarly journals Stability of limit cycle in a delayed model for tumor immune system competition with negative immune response

2006 ◽  
Vol 2006 ◽  
pp. 1-13 ◽  
Author(s):  
Radouane Yafia
Keyword(s):  
Immune Response ◽  
Immune System ◽  
Hopf Bifurcation ◽  
Discrete Delay ◽  
Local Asymptotic Stability ◽  
Nonlinear Delay Differential Equations ◽  
The Stability ◽  
Delay Differential ◽  
Delayed Model ◽  
Nonlinear Delay

This paper is devoted to the study of the stability of limit cycles of a system of nonlinear delay differential equations with a discrete delay. The system arises from a model of population dynamics describing the competition between tumor and immune system with negative immune response. We study the local asymptotic stability of the unique nontrivial equilibrium of the delay equation and we show that its stability can be lost through a Hopf bifurcation. We establish an explicit algorithm for determining the direction of the Hopf bifurcation and the stability or instability of the bifurcating branch of periodic solutions, using the methods presented by Diekmann et al.

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