scholarly journals Hopf Bifurcation of a Mathematical Model for Growth of Tumors with an Action of Inhibitor and Two Time Delays

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Bao Shi ◽  
Fangwei Zhang ◽  
Shihe Xu
Keyword(s):  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Time Delays ◽  
Cell Loss ◽  
Bifurcation Parameter ◽  
Discrete Delays

A mathematical model for growth of tumors with two discrete delays is studied. The delays, respectively, represent the time taken for cells to undergo mitosis and the time taken for the cell to modify the rate of cell loss due to apoptosis and kill of cells by the inhibitor. We show the influence of time delays on the Hopf bifurcation when one of delays is used as a bifurcation parameter.

On the Existence of Periodic Solutions of a Three-Patch Diffusion Predator-Prey System

2009 ◽  
Vol 64 (7-8) ◽  
pp. 405-410
Author(s):  
Mohammed Ismail ◽  
Atta A. K. Abu Hany ◽  
Aysha Agha
Keyword(s):  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Periodic Orbit ◽  
Time Delays ◽  
Positive Equilibrium ◽  
Bifurcation Parameter ◽  
Predator Prey ◽  
Prey System ◽  
Existence Of Periodic Solutions

AbstractWe establish a mathematical model for the three-patch diffusion predator-prey system with time delays. The theory of Hopf bifurcation is implemented, choosing the time delay parameter as a bifurcation parameter. We present the condition for the existence of a periodic orbit of the Hopf-type from the positive equilibrium.

scholarly journals Bifurcation of a Cohen-Grossberg Neural Network with Discrete Delays

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Qiming Liu ◽  
Wang Zheng
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Sufficient Conditions ◽  
Hopf Bifurcations ◽  
Bifurcation Parameter ◽  
Explicit Formulas ◽  
Global Hopf Bifurcation ◽  
Discrete Delays ◽  
Bifurcation And Stability

A simple Cohen-Grossberg neural network with discrete delays is investigated in this paper. The existence of local Hopf bifurcations is first considered by choosing the appropriate bifurcation parameter, and then explicit formulas are given to determine the direction of Hopf bifurcation and stability of the periodic solutions. Moreover, a set of sufficient conditions are given to guarantee the global Hopf bifurcation. Numerical simulations are given to illustrate the obtained results.

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.

Delay-Induced Bifurcation in High-Order Fractional Goodwin Models with Disparate Orders

2019 ◽  
Vol 29 (07) ◽  
pp. 1950090 ◽  
Author(s):  
Lingzhi Zhao ◽  
Chengdai Huang ◽  
Jinde Cao ◽  
Min Xiao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delays ◽  
Critical Value ◽  
High Order ◽  
High Dimensional ◽  
Bifurcation Parameter ◽  
Characteristic Roots ◽  
Goodwin Model ◽  
Linearized System

In the present paper, we first attempt to investigate the bifurcation for the delayed high-dimensional fractional Goodwin model with different orders. By taking the sum of time delays as a bifurcation parameter, the distribution of the characteristic roots of the linearized system is analyzed and the conditions of Hopf bifurcation are obtained. It is revealed that the model undergoes the Hopf bifurcation when the bifurcation parameter increases and exceeds the critical value. To verify the efficiency of our analytic findings, two simulation examples have been ultimately provided.

scholarly journals Dynamics and Stability Analysis of a Brucellosis Model with Two Discrete Delays

2018 ◽  
Vol 2018 ◽  
pp. 1-20 ◽  
Author(s):  
Paride O. Lolika ◽  
Steady Mushayabasa
Keyword(s):  
Mathematical Model ◽  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Incubation Period ◽  
Steady States ◽  
Discrete Delays ◽  
Two Delays

We present a mathematical model for brucellosis transmission that incorporates two discrete delays and culling of infected animals displaying signs of brucellosis infection. The first delay represents the incubation period while the second account for the time needed to detect and cull infectious animals. Feasibility and stability of the model steady states have been determined analytically and numerically. Further, the occurrence of Hopf bifurcation has been established. Overall the findings from the study, both analytical and numerical, suggest that the two delays can destabilize the system and periodic solutions can arise through Hopf bifurcation.

scholarly journals Modelling immune system dynamics: the interaction of HIV and recombinant virus

2015 ◽  
Vol 56 ◽  
pp. 30-35
Author(s):  
Ugnė Jankauskaitė ◽  
Olga Štikonienė
Keyword(s):  
Mathematical Model ◽  
Immune System ◽  
Numerical Simulations ◽  
Recombinant Virus ◽  
Time Delays ◽  
Dynamical Behavior ◽  
Hopf Bifurcations ◽  
Stability Switches ◽  
Discrete Delays ◽  
Theoretical Results

We investigate the dynamical behavior of a mathematical model of HIV and recombinant rabies virus (RV), designed to infect only the lymphocytes previously infected by HIV. This model is described by five ordinary differential equations with two discrete delays. The effect of two time delays on stability of the equilibria of the system has been studied. Stability switches and Hopf bifurcations when time delays cross through some critical values are found. Numerical simulations are performed to illustrate the theoretical results.

scholarly journals Hopf Bifurcation of a Differential-Algebraic Bioeconomic Model with Time Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Xiaojian Zhou ◽  
Xin Chen ◽  
Yongzhong Song
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Time Delays ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Bifurcation Parameter ◽  
Bioeconomic Model ◽  
Numerical Tests ◽  
The Stability

We investigate the dynamics of a differential-algebraic bioeconomic model with two time delays. Regarding time delay as a bifurcation parameter, we show that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Using the theories of normal form and center manifold, we also give the explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. Numerical tests are provided to verify our theoretical analysis.

HOPF BIFURCATION OF A TWO-NEURON NETWORK WITH DIFFERENT DISCRETE TIME DELAYS

2005 ◽  
Vol 15 (05) ◽  
pp. 1589-1601 ◽  
Author(s):  
SHAOWEN LI ◽  
XIAOFENG LIAO ◽  
CHUNGUANG LI ◽  
KWOK-WO WONG
Keyword(s):  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Periodic Solutions ◽  
Discrete Time ◽  
Time Delays ◽  
Bifurcation Parameter ◽  
Neuron Network ◽  
Bifurcation Theorem ◽  
Nyquist Criterion ◽  
Delay Dependent

In this paper, a two-neuron network with different time delays is investigated. By analyzing the associated characteristic equation, we obtain the conditions for delay-dependent and delay-independent asymptotic stability, respectively. Furthermore, we find that if the delay is used as a bifurcation parameter, Hopf bifurcation would occur. The direction and stability of the bifurcating periodic solutions are determined by using the Nyquist criterion and the graphical Hopf bifurcation theorem. Some examples are included to illustrate our results.

Oscillatory Dynamics of P53 Network With Time Delays

2018 ◽  
Vol 21 (6) ◽  
pp. 411-419 ◽  
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.

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