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.

scholarly journals Dynamics in a delayed diffusive cell cycle model

2018 ◽  
Vol 23 (5) ◽  
pp. 691-709
Author(s):  
Yanqin Wang ◽  
Ling Yang ◽  
Jie Yan
Keyword(s):  
Cell Cycle ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Spatial Dynamics ◽  
Positive Equilibrium ◽  
Cycle Model ◽  
Center Manifold Theorem ◽  
Diffusive Model ◽  
The Stability ◽  
Theoretical Results

In this paper, we construct a delayed diffusive model to explore the spatial dynamics of cell cycle in G2/M transition. We first obtain the local stability of the unique positive equilibrium for this model, which is irrelevant to the diffusion. Then, through investigating the delay-induced Hopf bifurcation in this model, we establish the existence of spatially homogeneous and inhomogeneous bifurcating periodic solutions. Applying the normal form and center manifold theorem of functional partial differential equations, we also determine the stability and direction of these bifurcating periodic solutions. Finally, numerical simulations are presented to validate our theoretical results.

Bifurcation Analysis of Phytoplankton and Zooplankton Interaction System with Two Delays

2020 ◽  
Vol 30 (03) ◽  
pp. 2050039
Author(s):  
Zhichao Jiang ◽  
Jiangtao Dai ◽  
Tongqian Zhang
Keyword(s):  
Periodic Solutions ◽  
Center Manifold ◽  
Threshold Values ◽  
Center Manifold Theory ◽  
Local Hopf Bifurcation ◽  
Two Delays ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results ◽  
Positivity And Boundedness

In this paper, the system of describing the interactions between poisonous phytoplankton and zooplankton is presented. It focuses on the effects of two delays on the dynamic behavior of the system. At first, the properties of solutions including positivity and boundedness are given. Next, the stability of equilibria and the existence of local Hopf bifurcation are established when delays change and cross some threshold values. Especially, the existence of global periodic solutions is discussed when the two delays are equal. Furthermore, the implicit algorithm is derived for deciding the properties of the branching periodic solutions by using center manifold theory. Some numerical simulations are performed for supporting the theoretical results. Finally, some conclusions are given.

scholarly journals Modeling and Mathematical Analysis of Labor Force Evolution

2019 ◽  
Vol 2019 ◽  
pp. 1-5
Author(s):  
Sanaa ElFadily ◽  
Abdelilah Kaddar
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Labor Force ◽  
Periodic Solutions ◽  
Mathematical Analysis ◽  
Equilibrium Position ◽  
Positive Equilibrium ◽  
Local Behavior ◽  
Bifurcation Theorem ◽  
Theoretical Results

In this paper, we propose a delayed differential system to model labor force (occupied labor force and unemployed) evolution. The mathematical analysis of our model focuses on the local behavior of the labor force around a positive equilibrium position; the existence of a branch of periodic solutions bifurcated from the positive equilibrium is then analyzed according to the Hopf bifurcation theorem. Finally, we performed numerical simulations to illustrate our theoretical results.

GLOBAL EXISTENCE OF PERIODIC SOLUTIONS IN THE LINEARLY COUPLED MACKEY–GLASS SYSTEM

2011 ◽  
Vol 21 (03) ◽  
pp. 711-724 ◽  
Author(s):  
YANQIU LI ◽  
WEIHUA JIANG
Keyword(s):  
Hopf Bifurcation ◽  
Global Existence ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Glass System ◽  
Positive Equilibrium ◽  
Higher Dimensional ◽  
Distribution Of Eigenvalues ◽  
Existence Of Periodic Solutions ◽  
The Stability

The dynamics of a linearly coupled Mackey–Glass system with delay are investigated. Based on the distribution of eigenvalues, we prove that a sequence of Hopf bifurcation occurs at the positive equilibrium as the delay increases and obtain the bifurcation set in the parameter plane. The explicit algorithm for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived, using the theories of normal form and center manifold. The global existence of periodic solutions is established using a global Hopf bifurcation result due to Wu [1998] and a Bendixson's criterion for higher dimensional ordinary differential equations due to [Li & Muldowney, 1993].

scholarly journals Dynamics Induced by Delay in a Nutrient-Phytoplankton Model with Multiple Delays

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-16 ◽  
Author(s):  
Chuanjun Dai ◽  
Hengguo Yu ◽  
Qing Guo ◽  
He Liu ◽  
Qi Wang ◽  
...  
Keyword(s):  
Numerical Simulations ◽  
Periodic Solutions ◽  
Positive Equilibrium ◽  
Multiple Delays ◽  
Asymptotically Stable ◽  
Phytoplankton Dynamics ◽  
Stability Switches ◽  
Globally Asymptotically Stable ◽  
Analytical Analysis ◽  
The Stability

A nutrient-phytoplankton model with multiple delays is studied analytically and numerically. The aim of this paper is to study how the delay factors influence dynamics of interaction between nutrient and phytoplankton. The analytical analysis indicates that the positive equilibrium is always globally asymptotically stable when the delay does not exist. On the contrary, the positive equilibrium loses its stability via Hopf instability induced by delay and then the corresponding periodic solutions emerge. Especially, the stability switches for positive equilibrium occur as the delay is increased. Furthermore, the numerical simulations show that periodic-2 and periodic-3 solutions can appear due to the existence of delays. Numerical results are consistent with the analytical results. Our results demonstrate that the delay has a great impact on the nutrient-phytoplankton dynamics.

HOPF BIFURCATION OF A DELAYED DIFFERENTIAL EQUATION

2007 ◽  
Vol 17 (04) ◽  
pp. 1367-1374 ◽  
Author(s):  
QIAN GUO ◽  
CHANGPIN LI
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Nonlinear Differential Equation ◽  
Trivial Solution ◽  
The Stability ◽  
Analytical Expressions ◽  
Theoretical Results

In this paper, we study Hopf bifurcation of a second-order nonlinear differential equation with time delay by using the Lyapunov–Schmidt reduction. The approximate analytical expressions of the periodic solutions bifurcated from the trivial solution are given. We also discuss the stability of these periodic solutions. The numerical simulations line up with the theoretical results.

scholarly journals Dynamics of a Delayed Model for the Transmission of Malicious Objects in Computer Network

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Periodic Solutions ◽  
Center Manifold ◽  
Computer Network ◽  
Center Manifold Theory ◽  
Two Delays ◽  
Normal Form Method ◽  
The Stability ◽  
Manifold Theory ◽  
Delayed Model ◽  
Theoretical Results

An SEIQRS model for the transmission of malicious objects in computer network with two delays is investigated in this paper. We show that possible combination of the two delays can affect the stability of the model and make the model bifurcate periodic solutions under some certain conditions. For further investigation, properties of the periodic solutions are studied by using the normal form method and center manifold theory. Finally, some numerical simulations are given to justify the theoretical results.

HOPF BIFURCATION ANALYSIS IN A MACKEY–GLASS SYSTEM

2007 ◽  
Vol 17 (06) ◽  
pp. 2149-2157 ◽  
Author(s):  
JUNJIE WEI ◽  
DEJUN FAN
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Higher Dimensional ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Equation With Delay ◽  
Bendixson Criterion

The dynamics of a Mackey–Glass equation with delay are investigated. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result due to Wu [1998] and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney [1994].

Stability and Bifurcation in a State-Dependent Delayed Predator–Prey System

2016 ◽  
Vol 26 (04) ◽  
pp. 1650060 ◽  
Author(s):  
Aiyu Hou ◽  
Shangjiang Guo
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Local Stability ◽  
Perturbation Methods ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Prey System ◽  
State Dependent ◽  
The Stability ◽  
Theoretical Results

In this paper, we consider a class of predator–prey equations with state-dependent delayed feedback. Firstly, we investigate the local stability of the positive equilibrium and the existence of the Hopf bifurcation. Then we use perturbation methods to determine the sub/supercriticality of Hopf bifurcation and hence the stability of Hopf bifurcating periodic solutions. Finally, numerical simulations supporting our theoretical results are also provided.

On the Stability Properties of a Periodically Forced, Time-Varying Mass System

2009 ◽  
Author(s):  
W. T. van Horssen ◽  
O. V. Pischanskyy ◽  
J. L. A. Dubbeldam
Keyword(s):  
Periodic Solutions ◽  
Forced Vibrations ◽  
Time Varying ◽  
Mass System ◽  
Single Degree Of Freedom ◽  
Stability Properties ◽  
Single Degree ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Varying Mass

In this paper the forced vibrations of a linear, single degree of freedom oscillator (sdofo) with a time-varying mass will be studied. The forced vibrations are due to small masses which are periodically hitting and leaving the oscillator with different velocities. Since these small masses stay for some time on the oscillator surface the effective mass of the oscillator will periodically vary in time. Not only solutions of the oscillator equation will be constructed, but also the stability properties, and the existence of periodic solutions will be discussed.

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