A model of a fishery with fish stock involving delay equations

2009 ◽  
Vol 367 (1908) ◽  
pp. 4907-4922 ◽  
Author(s):  
P. Auger ◽  
Arnaud Ducrot
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Delay Differential Equation ◽  
Bifurcation Analysis ◽  
Infinite Delay ◽  
Steady States ◽  
Delay Equations ◽  
Fish Stock ◽  
Delay Differential

The aim of this paper is to provide a new mathematical model for a fishery by including a stock variable for the resource. This model takes the form of an infinite delay differential equation. It is mathematically studied and a bifurcation analysis of the steady states is fulfilled. Depending on the different parameters of the problem, we show that Hopf bifurcation may occur leading to oscillating behaviours of the system. The mathematical results are finally discussed.

scholarly journals ADVANCED-DELAY DIFFERENTIAL EQUATION FOR AEROELASTIC OSCILLATIONS IN PHYSIOLOGY

2008 ◽  
Vol 03 (01n02) ◽  
pp. 125-133 ◽  
Author(s):  
JORGE C. LUCERO
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Delay Differential Equation ◽  
Oscillation Frequency ◽  
Vocal Folds ◽  
Flow Pressure ◽  
Functional Differential ◽  
Energy Dissipated ◽  
Delay Differential

This article analyzes a mathematical model for some aeroelastic oscillators in physiology, based on a previous representation for the vocal folds at phonation. The model characterizes the oscillation as superficial wave propagating through the tissues in the direction of the flow, and consists of a functional differential equation with advanced and delay arguments. The analysis shows that the oscillation occurs at a Hopf bifurcation, at which the energy absorbed from the flow overcomes the energy dissipated in the tissues. The bifurcation value of the flow pressure increases linearly with the tissue damping and the oscillation frequency. Also, it is minimum when the phase delay of the superficial wave to travel along the tissues is π, and increases indefinitely when the delay tends to 0 and to 2π.

Analytical Study of the Existence of a Hopf Bifurcation in the Tumor Cell Growth Model with Time Delay

2021 ◽  
Vol 3 (1) ◽  
pp. 20-28
Author(s):  
A. Yusnaeni ◽  
Kasbawati Kasbawati ◽  
Toaha Syamsuddin
Keyword(s):  
Differential Equation ◽  
Steady State ◽  
Immune System ◽  
Hopf Bifurcation ◽  
Delay Differential Equation ◽  
Immune Cells ◽  
Steady State Solution ◽  
State Solution ◽  
Activation Process ◽  
Delay Differential

AbstractIn this paper, we study a mathematical model of an immune response system consisting of a number of immune cells that work together to protect the human body from invading tumor cells. The delay differential equation is used to model the immune system caused by a natural delay in the activation process of immune cells. Analytical studies are focused on finding conditions in which the system undergoes changes in stability near a tumor-free steady-state solution. We found that the existence of a tumor-free steady-state solution was warranted when the number of activated effector cells was sufficiently high. By considering the lag of stimulation of helper cell production as the bifurcation parameter, a critical lag is obtained that determines the threshold of the stability change of the tumor-free steady state. It is also leading the system undergoes a Hopf bifurcation to periodic solutions at the tumor-free steady-state solution.Keywords: tumor–immune system; delay differential equation; transcendental function; Hopf bifurcation. AbstrakDalam makalah ini, dikaji model matematika dari sistem respon imun yang terdiri dari sejumlah sel imun yang bekerja sama untuk melindungi tubuh manusia dari invasi sel tumor. Persamaan diferensial tunda digunakan untuk memodelkan sistem kekebalan yang disebabkan oleh keterlambatan alami dalam proses aktivasi sel-sel imun. Studi analitik difokuskan untuk menemukan kondisi di mana sistem mengalami perubahan stabilitas di sekitar solusi kesetimbangan bebas tumor. Diperoleh bahwa solusi kesetimbangan bebas tumor dijamin ada ketika jumlah sel efektor yang diaktifkan cukup tinggi. Dengan mempertimbangkan tundaan stimulasi produksi sel helper sebagai parameter bifurkasi, didapatkan lag kritis yang menentukan ambang batas perubahan stabilitas dari solusi kesetimbangan bebas tumor. Parameter tersebut juga mengakibatkan sistem mengalami percabangan Hopf untuk solusi periodik pada solusi kesetimbangan bebas tumor.Kata kunci: sistem tumor–imun; persamaan differensial tundaan; fungsi transedental; bifurkasi Hopf.

τ-D Decomposition to the One Dimensional Delay Differential Equation by Fixed the Coefficient b<0

2013 ◽  
Vol 785-786 ◽  
pp. 1418-1422
Author(s):  
Ai Gao
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Delay Differential Equation ◽  
Stability Domain ◽  
Transcendental Equation ◽  
One Dimension ◽  
One Dimensional ◽  
The Stability ◽  
The One ◽  
Delay Differential

In this paper, we provide a partition of the roots of a class of transcendental equation by using τ-D decomposition ,where τ>0,a>0,b<0 and the coefficient b is fixed.According to the partition, one can determine the stability domain of the equilibrium and get a Hopf bifurcation diagram that can provide the Hopf bifurcation curves in the-parameter space, for one dimension delay differential equation .

scholarly journals Stability and Bifurcation Analysis for a Delay Differential Equation of Hepatitis B Virus Infection

2013 ◽  
Vol 2013 ◽  
pp. 1-15
Author(s):  
Xinchao Yang ◽  
Xiju Zong ◽  
Xingong Cheng ◽  
Zhenlai Han
Keyword(s):  
Differential Equation ◽  
Hepatitis B Virus ◽  
Virus Infection ◽  
Hepatitis B ◽  
Delay Differential Equation ◽  
Bifurcation Analysis ◽  
Hepatitis B Virus Infection ◽  
B Virus ◽  
Stability And Bifurcation Analysis ◽  
Delay Differential

The stability and bifurcation analysis for a delay differential equation of hepatitis B virus infection is investigated. We show the existence of nonnegative equilibria under some appropriated conditions. The existence of the Hopf bifurcation with delayτat the endemic equilibria is established by analyzing the distribution of the characteristic values. The explicit formulae which determine the direction of the bifurcations, stability, and the other properties of the bifurcating periodic solutions are given by using the normal form theory and the center manifold theorem. Numerical simulation verifies the theoretical results.

Supercritical and Subcritical Hopf Bifurcations in a Delay Differential Equation Model of a Heat-Exchanger Tube Under Cross-Flow

2019 ◽  
Vol 15 (2) ◽  
Author(s):  
Varun Vourganti ◽  
Shanti Swaroop Kandala ◽  
Vamsi C. Meesala ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equation ◽  
Heat Exchanger ◽  
Hopf Bifurcation ◽  
Delay Differential Equation ◽  
Cross Flow ◽  
Axial Loads ◽  
Galerkin Approximations ◽  
Subcritical Hopf Bifurcation ◽  
Delay Differential ◽  
Exchanger Tube

Abstract Nonlinear vibrations of a heat-exchanger tube modeled as a simply supported Euler–Bernoulli beam under axial load and cross-flow have been studied. The compressive axial loads are a consequence of thermal expansion, and tensile axial loads can be induced by design (prestress). The fluid forces are represented using an added mass, damping, and a time-delayed displacement term. Due to the presence of the time-delayed term, the equation governing the dynamics of the tube becomes a partial delay differential equation (PDDE). Using the modal-expansion procedure, the PDDE is converted into a nonlinear delay differential equation (DDE). The fixed points (zero and buckled equilibria) of the nonlinear DDE are found, and their linear stability is analyzed. It is found that stability can be lost via either supercritical or subcritical Hopf bifurcation. Using Galerkin approximations, the characteristic roots (spectrum) of the DDE are found and reported in the parametric space of fluid velocity and axial load. Furthermore, the stability chart obtained from the Galerkin approximations is compared with the critical curves obtained from analytical calculations. Next, the method of multiple scales (MMS) is used to derive the normal-form equations near the supercritical and subcritical Hopf bifurcation points for both zero and buckled equilibrium configurations. The steady-state amplitude response equation, obtained from the MMS, at Hopf bifurcation points is compared with the numerical solution. The coexistence of multiple limit cycles in the parametric space is found, and has implications in the fatigue life calculations of the heat-exchanger tubes.

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