Hopf-bifurcation Limit Cycles of an Extended Rosenzweig-MacArthur Model

2016 ◽  
Vol 8 (3) ◽  
pp. 22
Author(s):  
Eno E. Joshua ◽  
Chinwendu E. Madubueze ◽  
Ekemini T. Akpan
Keyword(s):  
Hopf Bifurcation ◽  
Phase Space ◽  
Limit Cycles ◽  
Local Stability ◽  
Bifurcation Parameter ◽  
Stability Criteria

<p>In this paper, we formulated a new topologically equivalence dynamics of an Extended Rosenzweig-MacArthur Model. Also, we investigated the local stability criteria, and determine the existence of co-dimension-1 Hopf-bifurcation limit cycles as the bifurcation-parameter changes. We discussed the dynamical complexities of this model using numerical responses, solution curves and phase-space diagrams.</p>

scholarly journals Hopf-bifurcation Limit Cycles of an Extended Rosenzweig-MacArthur Model

2016 ◽  
Vol 8 (3) ◽  
pp. 22
Author(s):  
Enobong E. Joshua ◽  
Ekemini T. Akpan ◽  
Chinwendu E. Madubueze
Keyword(s):  
Hopf Bifurcation ◽  
Phase Space ◽  
Limit Cycles ◽  
Local Stability ◽  
Bifurcation Parameter ◽  
Stability Criteria

<p>In this paper, we formulated a new topologically equivalence dynamics of an Extended Rosenzweig-MacArthur Model. Also, we investigated the local stability criteria, and determine the existence of co-dimension-1 Hopf-bifurcation limit cycles as the bifurcation-parameter changes. We discussed the dynamical complexities of this model using numerical responses, solution curves and phase-space diagrams.</p>

BIFURCATION ANALYSIS OF A DETRITUS-BASED ECOSYSTEM WITH TIME DELAY

2000 ◽  
Vol 08 (03) ◽  
pp. 255-261 ◽  
Author(s):  
DEBASIS MUKHERJEE ◽  
SANTANU RAY ◽  
DILIP KUMAR SINHA
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Bifurcation Analysis ◽  
Local Stability ◽  
Mangrove Ecosystem ◽  
Stability Criteria ◽  
Delay Effect

This article concentrates on the study of delay effect of a mangrove ecosystem of detritus, detritivores and predator of detritivores. Local stability criteria are derived in the absence of delays. Conditions are found out for which the system undergoes a Hopf bifurcation. Further conditions are derived for which there can be no change in stability.

QUALITATIVE ANALYSIS FOR THE MERKIN ENZYME REACTION SYSTEM

2002 ◽  
Vol 10 (02) ◽  
pp. 167-182
Author(s):  
YUQUAN WANG ◽  
ZUORUI SHEN
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Local Stability ◽  
Bifurcation Theory ◽  
Sufficient Conditions ◽  
Reaction System ◽  
Enzyme Reaction ◽  
Qualitative Theory ◽  
Positive Equilibrium ◽  
Hopf Bifurcations

Applying qualitative theory and Hopf bifurcation theory, we detailedly discuss the Merkin enzyme reaction system, and the sufficient conditions derived for the global stability of the unique positive equilibrium, the local stability of three equilibria and the existence of limit cycles. Meanwhile, we show that the Hopf bifurcations may occur. Using MATLAB software, we present three examples to simulate these conclusions in this paper.

A Competition Chemostat with General Variable Yield and Growth Rates in the Presence of a Internal Inhibitor

2011 ◽  
Vol 110-116 ◽  
pp. 285-293
Author(s):  
Jian Mei Luo ◽  
Yu Ming Feng
Keyword(s):  
Hopf Bifurcation ◽  
Qualitative Analysis ◽  
Limit Cycles ◽  
Growth Rates ◽  
Local Stability ◽  
Nutrient Concentration ◽  
The Other ◽  
Global Asymptotical Stability ◽  
Variable Yield ◽  
Increasing Function

This paper considered two organisms competing for a nutrient in the chemostat in the presence of an inhibitor, where the yields and growth rates are general increasing function of the nutrient concentration. The inhibitor is produced by one organisms and is lethal to the other organism. By the theory of qualitative analysis for ordinary equations, first, conditions of the existence and local stability of the rest points are obtain; then the global asymptotical stability, the existence of limit cycles and Hopf bifurcation are discussed.

scholarly journals HOPF BIFURCATION ANALYSIS FOR A MATHEMATICAL MODEL OF P53-MDM2 INTERACTION

2008 ◽  
Vol 18 (01) ◽  
pp. 275-283 ◽  
Author(s):  
MIHAELA NEAMŢU ◽  
RAUL FLORIN HORHAT ◽  
DUMITRU OPRIŞ
Keyword(s):  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Stationary State ◽  
Bifurcation Analysis ◽  
Local Stability ◽  
Bifurcation Parameter ◽  
Simple Mathematical Model

In this paper we analyze a simple mathematical model which describes the interaction between proteins P53 and Mdm2. For the stationary state we discuss the local stability and the existence of a Hopf bifurcation. We study the direction and stability of the bifurcating periodic solutions by choosing the delay as a bifurcation parameter. Finally, we will offer some numerical simulations and present our conclusions.

scholarly journals INSTABILITIES, POINT ATTRACTORS AND LIMIT CYCLES IN AN INFLATIONARY UNIVERSE

1995 ◽  
Vol 10 (40) ◽  
pp. 3119-3127 ◽  
Author(s):  
LUCA SALASNICH
Keyword(s):  
Phase Space ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Inflationary Universe ◽  
Bifurcation Parameter ◽  
Inflaton Field ◽  
The Stability

We study the stability of a scalar inflaton field and analyze its point attractors in the phase space. We show that the value of the inflaton field in the vacuum is a bifurcation parameter and prove the possible existence of a limit cycle by using analytical and numerical arguments.

Stability and multi-parametric Hopf bifurcation analyses of viral infection model with time delay

2015 ◽  
Vol 08 (05) ◽  
pp. 1550059 ◽  
Author(s):  
M. Prakash ◽  
P. Balasubramaniam
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Viral Infection ◽  
Limit Cycles ◽  
Vital Role ◽  
Logistic Growth ◽  
Infection Model ◽  
Bifurcation Parameter ◽  
Delay Differential ◽  
Manifold Theory

Ever since HIV was first diagnosed in human, a great number of scientific works have been undertaken to explore the biological mechanisms involved in the infection and progression of the disease. This paper deals with stability and bifurcation analyses of mathematical model that represents the dynamics of HIV infection of thymus. The existence and stability of the equilibria are investigated. The model is described by a system of delay differential equations with logistic growth term, cure rate and discrete type of time delay. Choosing the time delay as a bifurcation parameter, the analysis is mainly focused on the Hopf bifurcation problem to predict the existence of a limit cycle bifurcating from the infected steady state. Further, using center manifold theory and normal form method we derive explicit formulae to determine the stability and direction of the limit cycles. Moreover the mitosis rate r also plays a vital role in the model, so we fix it as second bifurcation parameter in the incidence of viral infection. Our analysis shows that, while both the bifurcation parameters can destabilize the equilibrium E* and cause limit cycles. Numerical simulations are performed to investigate the qualitative behaviors of the inherent model.

scholarly journals A time-delay COVID-19 propagation model considering supply chain transmission and hierarchical quarantine rate

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Fangfang Yang ◽  
Zizhen Zhang
Keyword(s):  
Supply Chain ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Local Stability ◽  
Propagation Model ◽  
Bifurcation Parameter ◽  
Two Delays ◽  
Chain Transmission ◽  
Theoretical Results

AbstractIn this manuscript, we investigate a novel Susceptible–Exposed–Infected–Quarantined–Recovered (SEIQR) COVID-19 propagation model with two delays, and we also consider supply chain transmission and hierarchical quarantine rate in this model. Firstly, we analyze the existence of an equilibrium, including a virus-free equilibrium and a virus-existence equilibrium. Then local stability and the occurrence of Hopf bifurcation have been researched by thinking of time delay as the bifurcation parameter. Besides, we calculate direction and stability of the Hopf bifurcation. Finally, we carry out some numerical simulations to prove the validity of theoretical results.

scholarly journals Dynamical analysis of a giving up smoking model with time delay

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zizhen Zhang ◽  
Ruibin Wei ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory

AbstractIn this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

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