scholarly journals Periodic Oscillations in a Chemostat Model with Two Discrete Delays

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Tiansi Zhang ◽  
Xuehui Ji ◽  
Bo Li
Keyword(s):  
Nutrient Recycling ◽  
Type Model ◽  
Positive Equilibrium ◽  
Chemostat Model ◽  
Periodic Oscillations ◽  
System A ◽  
Limiting Nutrient ◽  
Discrete Delays ◽  
Linearized System ◽  
Theoretical Results

Periodic oscillations of solutions of a chemostat-type model in which a species feeds on a limiting nutrient are considered. The model incorporates two discrete delays representing the lag in nutrient recycling and nutrient conversion. Through the study of characteristic equation associated with the linearized system, a unique positive equilibrium is found and proved to be locally asymptotically stable under some conditions. Meanwhile, a Hopf bifurcation causing periodic solutions is also obtained. Numerical simulations illustrate the theoretical results.

GLOBAL STABILITY OF A CHEMOSTAT MODEL WITH DELAYED RESPONSE IN GROWTH AND A LETHAL EXTERNAL INHIBITOR

2008 ◽  
Vol 01 (04) ◽  
pp. 503-520 ◽  
Author(s):  
ZHIQI LU ◽  
JINGJING WU
Keyword(s):  
Global Stability ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Delayed Response ◽  
Qualitative Properties ◽  
Chemostat Model ◽  
Stability And Instability ◽  
Discrete Delays ◽  
Globally Stable

A competition model between two species with a lethal inhibitor in a chemostat is analyzed. Discrete delays are used to describe the nutrient conversion process. The proved qualitative properties of the solution are positivity, boundedness. By analyzing the local stability of equilibria, it is found that the conditions for stability and instability of the boundary equilibria are similar to those in [9]. In addition, the global asymptotic behavior of the system is discussed and the sufficient conditions for the global stability of the boundary equilibria are obtained. Moreover, by numerical simulation, it is interesting to find that the positive equilibrium may be globally stable.

STABILITY ANALYSIS OF A RATIO-DEPENDENT CHEMOSTAT MODEL WITH TIME DELAY AND VARIABLE YIELD

2010 ◽  
Vol 03 (02) ◽  
pp. 243-253 ◽  
Author(s):  
ZHE LI ◽  
RUI XU
Keyword(s):  
Time Delay ◽  
Lyapunov Functional ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Chemostat Model ◽  
Variable Yield ◽  
Boundary Equilibrium ◽  
Comparison Arguments ◽  
Ratio Dependent ◽  
Theoretical Results

A chemostat model with time delay, variable yield and ratio-dependent functional response is investigated. By analyzing the corresponding characteristic equations, the local stability of a boundary equilibrium and a positive equilibrium is discussed and the existence of Hopf bifurcation is established. By using the comparison arguments, sufficient conditions are obtained for the global stability of the boundary equilibrium. By constructing a suitable Lyapunov functional, sufficient conditions are derived for the global asymptotic stability of the positive equilibrium. Finally, numerical simulations are carried out to illustrate the theoretical results.

scholarly journals Hopf Bifurcation Analysis for a Two Species Periodic Chemostat Model with Discrete Delays

2020 ◽  
pp. 93-105
Author(s):  
Jane Ireri ◽  
Ganesh Pokhariyal ◽  
Stephene Moindi
Keyword(s):  
Periodic Solution ◽  
Fourier Series ◽  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Nutrient Input ◽  
Chemostat Model ◽  
Bifurcation Theorem ◽  
Interior Equilibrium ◽  
Limiting Nutrient ◽  
Discrete Delays

In this paper we analyze a Chemostat model of two species competing for a single limiting nutrient input varied periodically using a Fourier series with discrete delays. To understand global aspects of the dynamics we use an extension of the Hopf bifurcation theorem, a method that rigorously establishes existence of a periodic solution. We show that the interior equilibrium point changes its stability and due to the delay parameter it undergoes a Hopf bifurcation.Numerical results shows that coexistence is possible when delays are introduced and Fourier series produces the required seasonal variations. We also show that for small delays periodic variations of nutrients has more influence on species density variations than the delay.

scholarly journals Stability and Bifurcation Analysis of a Nonlinear Discrete Logistic Model with Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Daiyong Wu ◽  
Hai Zhang ◽  
Jinde Cao ◽  
Tasawar Hayat
Keyword(s):  
Asymptotic Stability ◽  
Logistic Model ◽  
Bifurcation Theory ◽  
Polynomial Equation ◽  
Positive Equilibrium ◽  
Stability And Bifurcation Analysis ◽  
Linearized System ◽  
Bifurcation Direction ◽  
The Stability ◽  
Theoretical Results

We consider a nonlinear discrete logistic model with delay. The characteristic equation of the linearized system at the positive equilibrium is a polynomial equation involving high order terms. We obtain the conditions ensuring the asymptotic stability of the positive equilibrium and the existence of Neimark-Sacker bifurcation, with respect to the parameter of the model. Based on the bifurcation theory, we discuss Neimark-Sacker bifurcation direction and the stability of bifurcated solutions. Finally, some numerical simulations are performed to illustrate the theoretical results.

scholarly journals Would depositors pay to show that they do not withdraw? Theory and experiment

2020 ◽  
Vol 23 (3) ◽  
pp. 873-894
Author(s):  
Markus Kinateder ◽  
Hubert János Kiss ◽  
Ágnes Pintér
Keyword(s):  
Multiple Equilibria ◽  
Financial Intermediation ◽  
Type Model ◽  
Unique Equilibrium ◽  
Equilibrium Outcome ◽  
Bank Run ◽  
Public Announcements ◽  
Information Sets ◽  
Theoretical Results ◽  
Theory And Experiment

Abstract In a Diamond–Dybvig type model of financial intermediation, we allow depositors to announce at a positive cost to subsequent depositors that they keep their funds deposited in the bank. Theoretically, the mere availability of public announcements (and not its use) ensures that no bank run is the unique equilibrium outcome. Multiple equilibria—including bank run—exist without such public announcements. We test the theoretical results in the lab and find a widespread use of announcements, which we interpret as an attempt to coordinate on the no bank run outcome. Withdrawal rates in general are lower in information sets that contain announcements.

scholarly journals Asymptotic Stability of Caputo Type Fractional Neutral Dynamical Systems with Multiple Discrete Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Hai Zhang ◽  
Daiyong Wu ◽  
Jinde Cao
Keyword(s):  
Asymptotic Stability ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Algebraic Approach ◽  
Stability Criteria ◽  
Differential Systems ◽  
Discrete Delays ◽  
Transcendental Equations ◽  
The Stability ◽  
Theoretical Results

We discuss the delay-independent asymptotic stability of Caputo type fractional-order neutral differential systems with multiple discrete delays. Based on the algebraic approach and matrix theory, the sufficient conditions are derived to ensure the asymptotic stability for all time-delay parameters. By applying the stability criteria, one can avoid solving the roots of transcendental equations. The results obtained are computationally flexible and convenient. Moreover, an example is provided to illustrate the effectiveness and applicability of the proposed theoretical results.

scholarly journals Analysis of a diffusive cholera model incorporating latency and bacterial hyperinfectivity

2021 ◽  
Vol 20 (11) ◽  
pp. 3921
Author(s):  
Wei Yang ◽  
Jinliang Wang
Keyword(s):  
Lyapunov Functional ◽  
Global Attractivity ◽  
Reproduction Number ◽  
Reaction Diffusion ◽  
Threshold Dynamics ◽  
Positive Equilibrium ◽  
Flux Boundary Condition ◽  
Nonlocal Reaction ◽  
Theoretical Results ◽  
The Basic Reproduction Number

<p style='text-indent:20px;'>In this paper, we are concerned with the threshold dynamics of a diffusive cholera model incorporating latency and bacterial hyperinfectivity. Our model takes the form of spatially nonlocal reaction-diffusion system associated with zero-flux boundary condition and time delay. By studying the associated eigenvalue problem, we establish the threshold dynamics that determines whether or not cholera will spread. We also confirm that the threshold dynamics can be determined by the basic reproduction number. By constructing Lyapunov functional, we address the global attractivity of the unique positive equilibrium whenever it exists. The theoretical results are still hold for the case when the constant parameters are replaced by strictly positive and spatial dependent functions.</p>

scholarly journals Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1280
Author(s):  
Liyun Lai ◽  
Zhenliang Zhu ◽  
Fengde Chen
Keyword(s):  
Allee Effect ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Prey Model ◽  
Strong Allee Effect ◽  
Theoretical Results

We proposed and analyzed a predator–prey model with both the additive Allee effect and the fear effect in the prey. Firstly, we studied the existence and local stability of equilibria. Some sufficient conditions on the global stability of the positive equilibrium were established by applying the Dulac theorem. Those results indicate that some bifurcations occur. We then confirmed the occurrence of saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Those theoretical results were demonstrated with numerical simulations. In the bifurcation analysis, we only considered the effect of the strong Allee effect. Finally, we found that the stronger the fear effect, the smaller the density of predator species. However, the fear effect has no influence on the final density of the prey.

Qualitative analysis of a chemostat model with inhibitory exponential substrate uptake and a time delay

2014 ◽  
Vol 07 (04) ◽  
pp. 1450045 ◽  
Author(s):  
Qinglai Dong ◽  
Wanbiao Ma
Keyword(s):  
Time Delay ◽  
Asymptotic Stability ◽  
Qualitative Analysis ◽  
Numerical Simulations ◽  
Global Asymptotic Stability ◽  
Positive Equilibrium ◽  
Substrate Uptake ◽  
Sufficient Condition ◽  
Chemostat Model ◽  
Local Asymptotic Stability

In this paper, we consider a simple chemostat model with inhibitory exponential substrate uptake and a time delay. A detailed qualitative analysis about existence and boundedness of its solutions and the local asymptotic stability of its equilibria are carried out. Using Lyapunov–LaSalle invariance principle, we show that the washout equilibrium is global asymptotic stability for any time delay. Using the fluctuation lemma, the sufficient condition of the global asymptotic stability of the positive equilibrium [Formula: see text] is obtained. Numerical simulations are also performed to illustrate the results.

Sign in / Sign up

Export Citation Format

Share Document