scholarly journals Competition between Plasmid-Bearing and Plasmid-Free Organisms in a Chemostat with Pulsed Input and Washout

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Sanling Yuan ◽  
Yu Zhao ◽  
Anfeng Xiao
Keyword(s):  
Periodic Solution ◽  
Stability Analysis ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Bifurcation Theory ◽  
Periodic Oscillations ◽  
The Stability ◽  
Invasion Threshold ◽  
Standard Techniques

We consider a model of competition between plasmid-bearing and plasmid-free organisms in the chemostat with pulsed input and washout. We investigate the subsystem with nutrient and plasmid-free organism and study the stability of the boundary periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields the invasion threshold of the plasmid-bearing organism. By using the standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, plasmid-free, and plasmid-bearing organisms. Numerical simulations are carried out to illustrate our results.

ANALYSIS OF A MODEL OF PLASMID-BEARING, PLASMID-FREE COMPETITION IN A PULSED CHEMOSTAT

2006 ◽  
Vol 09 (03) ◽  
pp. 263-276 ◽  
Author(s):  
XIANGYUN SHI ◽  
XINYU SONG ◽  
XUEYONG ZHOU
Keyword(s):  
Periodic Solution ◽  
Stability Analysis ◽  
Bifurcation Theory ◽  
Continuous System ◽  
Impulsive Effect ◽  
Chemostat Model ◽  
Free Competition ◽  
The Stability ◽  
Invasion Threshold ◽  
Standard Techniques

We introduce and study a chemostat model with plasmid-bearing, plasmid-free competition and impulsive effect. According to the stability analysis of the boundary periodic solution, we obtain the invasion threshold of the plasmid-free organism and plasmid-bearing organism. Furthermore, by using standard techniques of bifurcation theory, we prove the system has a positive τ-periodic solution, which shows that the impulsive effect destroys the equilibria of the unforced continuous system and initiates the periodic solution. Our results can be applied to control bioreactors aimed at producing commercial products through genetically altered organisms.

scholarly journals Bifurcation Analysis of a Chemostat Model of Plasmid-Bearing and Plasmid-Free Competition with Pulsed Input

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Zhong Zhao ◽  
Baozhen Wang ◽  
Liuyong Pang ◽  
Ying Chen
Keyword(s):  
Periodic Solution ◽  
Bifurcation Analysis ◽  
Bifurcation Theory ◽  
Chemostat Model ◽  
Periodic Oscillations ◽  
Free Competition ◽  
The Stability ◽  
Invasion Threshold ◽  
Standard Techniques

A chemostat model of plasmid-bearing and plasmid-free competition with pulsed input is proposed. The invasion threshold of the plasmid-bearing and plasmid-free organisms is obtained according to the stability of the boundary periodic solution. By use of standard techniques of bifurcation theory, the periodic oscillations in substrate, plasmid-bearing, and plasmid-free organisms are shown when some conditions are satisfied. Our results can be applied to control bioreactor aimed at producing commercial producers through genetically altered organisms.

BIFURCATION OF A MUTUALISTIC SYSTEM WITH VARIABLE COEFFICIENTS AND IMPULSIVE EFFECTS

2009 ◽  
Vol 02 (03) ◽  
pp. 363-375 ◽  
Author(s):  
YONGZHEN PEI ◽  
YONG YANG ◽  
CHANGGUO LI
Keyword(s):  
Periodic Solution ◽  
Numerical Simulations ◽  
Bifurcation Theory ◽  
Positive Periodic Solution ◽  
Variable Coefficients ◽  
Impulsive Effects ◽  
The Stability ◽  
Invasion Threshold ◽  
Standard Techniques

In this paper, we introduce and study an impulsive mutualistic model with variable coefficients. By using Floquent theorem, the invasion threshold and the stability of the boundary periodic solution are obtained. Furthermore, by using standard techniques of bifurcation theory, the existent condition of the positive periodic solution is obtained. Finally, numerical simulations are carried out to confirm the main theorems.

scholarly journals On the dynamics of a vaccination model with multiple transmission ways

2013 ◽  
Vol 23 (4) ◽  
pp. 761-772 ◽  
Author(s):  
Shu Liao ◽  
Weiming Yang
Keyword(s):  
Stability Analysis ◽  
Disease Control ◽  
Numerical Simulations ◽  
Bifurcation Theory ◽  
Reproduction Number ◽  
Model Predictions ◽  
The Stability ◽  
Disease Free ◽  
Multiple Transmission

Abstract In this paper, we present a vaccination model with multiple transmission ways and derive the control reproduction number. The stability analysis of both the disease-free and endemic equilibria is carried out, and bifurcation theory is applied to explore a variety of dynamics of this model. In addition, we present numerical simulations to verify the model predictions. Mathematical results suggest that vaccination is helpful for disease control by decreasing the control reproduction number below unity.

scholarly journals Stability Analysis for Mutually Delay-Coupled Semiconductor Lasers System

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Rina Su ◽  
Chunrui Zhang ◽  
Zuolin Shen
Keyword(s):  
Stability Analysis ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Semiconductor Lasers ◽  
Hopf Bifurcations ◽  
Stability Switch ◽  
Multiple Periodic Solutions ◽  
The Stability ◽  
New Phenomena ◽  
Coupled Semiconductor

In this paper, the problem of the stability for mutually delay-coupled semiconductor lasers system is investigated. By analyzing the associated characteristic equation, linear stability is investigated and Hopf bifurcations are demonstrated. The new phenomena such as stability switch is found. TheZ2equivariant property and the existence of multiple periodic solutions is also discussed. Numerical simulations are presented to illustrate the results in the paper.

Graphical Hopf Bifurcation of a Filippov HTLV-1 Model With Delay in Cytotoxic T Cells Response

2018 ◽  
Vol 140 (9) ◽  
Author(s):  
Elham Shamsara ◽  
Zahra Afsharnezhad ◽  
Elham Javidmanesh
Keyword(s):  
T Cells ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Bifurcation Theory ◽  
Cytotoxic T Cells ◽  
Dynamical Behavior ◽  
Bifurcation Theorem ◽  
Nyquist Criterion ◽  
The Stability ◽  
Frequency Domain Approach

In this paper, we present a discontinuous cytotoxic T cells (CTLs) response for HTLV-1. Moreover, a delay parameter for the activation of CTLs is considered. In fact, a system of differential equation with discontinuous right-hand side with delay is defined for HTLV-1. For analyzing the dynamical behavior of the system, graphical Hopf bifurcation is used. In general, Hopf bifurcation theory will help to obtain the periodic solutions of a system as parameter varies. Therefore, by applying the frequency domain approach and analyzing the associated characteristic equation, the existence of Hopf bifurcation by using delay immune response as a bifurcation parameter is determined. The stability of Hopf bifurcation periodic solutions is obtained by the Nyquist criterion and the graphical Hopf bifurcation theorem. At the end, numerical simulations demonstrated our results for the system of HTLV-1.

scholarly journals The linear instability of the stratified plane Couette flow

2018 ◽  
Vol 853 ◽  
pp. 205-234 ◽  
Author(s):  
Giulio Facchini ◽  
Benjamin Favier ◽  
Patrice Le Gal ◽  
Meng Wang ◽  
Michael Le Bars
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Numerical Simulations ◽  
Couette Flow ◽  
Vertical Direction ◽  
Stability Diagram ◽  
Linear Instability ◽  
Plane Couette Flow ◽  
Horizontal Shear ◽  
The Stability

We present the stability analysis of a plane Couette flow which is stably stratified in the vertical direction orthogonal to the horizontal shear. Interest in such a flow comes from geophysical and astrophysical applications where background shear and vertical stable stratification commonly coexist. We perform the linear stability analysis of the flow in a domain which is periodic in the streamwise and vertical directions and confined in the cross-stream direction. The stability diagram is constructed as a function of the Reynolds number $Re$ and the Froude number $Fr$, which compares the importance of shear and stratification. We find that the flow becomes unstable when shear and stratification are of the same order (i.e. $Fr\sim 1$) and above a moderate value of the Reynolds number $Re\gtrsim 700$. The instability results from a wave resonance mechanism already known in the context of channel flows – for instance, unstratified plane Couette flow in the shallow-water approximation. The result is confirmed by fully nonlinear direct numerical simulations and, to the best of our knowledge, constitutes the first evidence of linear instability in a vertically stratified plane Couette flow. We also report the study of a laboratory flow generated by a transparent belt entrained by two vertical cylinders and immersed in a tank filled with salty water, linearly stratified in density. We observe the emergence of a robust spatio-temporal pattern close to the threshold values of $Fr$ and $Re$ indicated by linear analysis, and explore the accessible part of the stability diagram. With the support of numerical simulations we conclude that the observed pattern is a signature of the same instability predicted by the linear theory, although slightly modified due to streamwise confinement.

scholarly journals Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay

2021 ◽  
Vol 26 (3) ◽  
pp. 375-395
Author(s):  
Rina Su ◽  
Chunrui Zhang
Keyword(s):  
Normal Form ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Center Manifold Theorem ◽  
Distribution Of Eigenvalues ◽  
Normal Form Method ◽  
The Stability

In this paper, the Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay was explored. First, the conditions of the occurrence of Hopf-zero bifurcation were obtained by analyzing the distribution of eigenvalues in correspondence to linearization. Second, the stability of Hopf-zero bifurcation periodic solutions was determined based on the discussion of the normal form of the system, and some numerical simulations were employed to illustrate the results of this study. Lastly, the normal form of the system on the center manifold was derived by using the center manifold theorem and normal form method.

scholarly journals Periodic Peakon and Smooth Periodic Solutions for KP-MEW(3,2) Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Junning Cai ◽  
Minzhi Wei ◽  
Liping He
Keyword(s):  
Dynamical System ◽  
Periodic Solution ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Phase Portrait ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Wave System ◽  
Integral Constant ◽  
Straight Line

In this paper, we consider the KP-MEW(3,2) equation by the bifurcation theory of dynamical systems when integral constant is considered. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. The phase portrait for c < 0 , 0 < c < 1 , and c > 1 is drawn. Exact parametric representations of periodic peakon solutions and smooth periodic solution are presented.

Dynamic Analysis in a Plankton Ecosystem Involving Two Delays

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Zhichao Jiang ◽  
Weicong Zhang ◽  
Xueli Bai ◽  
Maoyan Jie
Keyword(s):  
Dynamic Analysis ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Positive Equilibrium ◽  
Switching Phenomenon ◽  
Wide Range ◽  
Two Delays ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Theoretical Results

In this paper, a phytoplankton and zooplankton relationship system with two delays is investigated whose coefficients are related to one of the two delays. Firstly, the dynamic behaviors of the system with one delay are given and the stability of positive equilibrium and the existence of periodic solutions are obtained. Using the fact that the system may occur, the stable switching phenomenon is verified. Under certain conditions, the periodic solutions will exist in a wide range as the delay gets away from critical values. Fixing the delay [Formula: see text] in the stable interval, it is revealed that the effect of [Formula: see text] can also cause the vibration of system. This explains that two delays play an important role in the oscillation behavior of the system. Furthermore, using the crossing curve methods, the stable changes of the positive equilibrium in two-delays plane are given, which generalizes the results of systems for which the coefficients do not depend on delay. Some numerical simulations are provided to verify the theoretical results.

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