scholarly journals Nonlinear Dynamics of a Nutrient-Phytoplankton Model with Time Delay

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
DeBing Mei ◽  
Min Zhao ◽  
Hengguo Yu ◽  
Chuanjun Dai ◽  
Yi Wang
Keyword(s):  
Time Delay ◽  
Center Manifold ◽  
Critical Value ◽  
Hopf Bifurcations ◽  
Type Ii ◽  
Center Manifold Theory ◽  
Existence And Stability ◽  
Manifold Theory ◽  
Theoretical Results ◽  
Type Ii Functional Response

We consider a nutrient-phytoplankton model with a Holling type II functional response and a time delay. By selecting the time delay used as a bifurcation parameter, we prove that the system is stable if the delay value is lower than the critical value but unstable when it is above this value. First, we investigate the existence and stability of the equilibria, as well as the existence of Hopf bifurcations. Second, we consider the direction, stability, and period of the periodic solutions from the steady state based on the normal form and the center manifold theory, thereby deriving explicit formulas. Finally, some numerical simulations are given to illustrate the main theoretical results.

Stability and Hopf Bifurcation of a Delayed Mutualistic System

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Ruimin Zhang ◽  
Xiaohui Liu ◽  
Chunjin Wei
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Stage Structure ◽  
Center Manifold Theory ◽  
Mutualistic Relationship ◽  
Normal Form Method ◽  
Explicit Formulae ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

In this paper, we study a classic mutualistic relationship between the leaf cutter ants and their fungus garden, establishing a time delay mutualistic system with stage structure. We investigate the stability and Hopf bifurcation by analyzing the distribution of the roots of the associated characteristic equation. By means of the center manifold theory and normal form method, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Finally, some numerical simulations are carried out for illustrating the theoretical results.

scholarly journals Stability and Hopf Bifurcation Analysis of a Nutrient-Phytoplankton Model with Delay Effect

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Xinhong Pan ◽  
Min Zhao ◽  
Chuanjun Dai ◽  
Yapei Wang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Center Manifold Theory ◽  
Delay Effect ◽  
Delay Differential System ◽  
The Stability ◽  
Delay Differential ◽  
Manifold Theory ◽  
Two Equilibria

A delay differential system is investigated based on a previously proposed nutrient-phytoplankton model. The time delay is regarded as a bifurcation parameter. Our aim is to determine how the time delay affects the system. First, we study the existence and local stability of two equilibria using the characteristic equation and identify the condition where a Hopf bifurcation can occur. Second, the formulae that determine the direction of the Hopf bifurcation and the stability of periodic solutions are obtained using the normal form and the center manifold theory. Furthermore, our main results are illustrated using numerical simulations.

scholarly journals Chaos and Hopf Bifurcation Analysis of the Delayed Local Lengyel-Epstein System

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qingsong Liu ◽  
Yiping Lin ◽  
Jingnan Cao ◽  
Jinde Cao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Critical Value ◽  
Hopf Bifurcations ◽  
The Stability

The local reaction-diffusion Lengyel-Epstein system with delay is investigated. By choosingτas bifurcating parameter, we show that Hopf bifurcations occur when time delay crosses a critical value. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to support the analytical results and the chaotic behaviors are observed.

scholarly journals Bifurcation and Patterns Analysis for a Spatiotemporal Discrete Gierer-Meinhardt System

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 243
Author(s):  
Biao Liu ◽  
Ranchao Wu
Keyword(s):  
Bifurcation Theory ◽  
Center Manifold ◽  
Turing Instability ◽  
Flip Bifurcation ◽  
Center Manifold Theory ◽  
Pattern Dynamics ◽  
Instability Theory ◽  
Pattern Formations ◽  
Manifold Theory ◽  
Theoretical Results

The Gierer-Meinhardt system is one of the prototypical pattern formation models. The bifurcation and pattern dynamics of a spatiotemporal discrete Gierer-Meinhardt system are investigated via the couple map lattice model (CML) method in this paper. The linear stability of the fixed points to such spatiotemporal discrete system is analyzed by stability theory. By using the bifurcation theory, the center manifold theory and the Turing instability theory, the Turing instability conditions in flip bifurcation and Neimark–Sacker bifurcation are considered, respectively. To illustrate the above theoretical results, numerical simulations are carried out, such as bifurcation diagram, maximum Lyapunov exponents, phase orbits, and pattern formations.

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.

STABILITY AND BIFURCATION ANALYSIS IN A DELAYED PREDATOR-PREY SYSTEM

2009 ◽  
Vol 02 (04) ◽  
pp. 483-506 ◽  
Author(s):  
ZHICHAO JIANG ◽  
WENZHI ZHANG ◽  
DONGSHENG HUO
Keyword(s):  
Local Stability ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Center Manifold Theory ◽  
Predator Prey ◽  
Prey System ◽  
Stability And Bifurcation Analysis ◽  
Characteristic Values ◽  
The Stability ◽  
Manifold Theory

A delayed ratio-dependent one-predator and two-prey system with Michaelis–Menten type functional response is investigated. We show the existence of nonnegative equilibria under some appropriated conditions. Criteria for local stability, instability of nonnegative equilibria are obtained. The existence of Hopf bifurcations at the endemic equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by using the normal form and the center manifold theory. At last, some numerical simulations to support the analytical conclusions are carried out.

Stability and Bifurcation Analysis in a Food Chain Model with Delay

2011 ◽  
Vol 110-116 ◽  
pp. 3382-3388
Author(s):  
Zhang Li
Keyword(s):  
Hopf Bifurcation ◽  
Food Chain ◽  
Center Manifold ◽  
Chain Model ◽  
Center Manifold Theory ◽  
Stability And Bifurcation Analysis ◽  
Existence And Stability ◽  
The Stability ◽  
Manifold Theory ◽  
Food Chain Model

In this paper, we investigate a delayed three-species food chain model. The existence and stability of equilibria are obtained. A explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory.

scholarly journals Hopf Bifurcation Analysis for the Modified Rayleigh Price Model with Time Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yanhui Zhai ◽  
Haiyun Bai ◽  
Ying Xiong ◽  
Xiaona Ma
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Center Manifold ◽  
Center Manifold Theory ◽  
Price Model ◽  
Stability Switches ◽  
Normal Form Theory ◽  
Form Theory ◽  
Manifold Theory

This paper mainly modifies and further develops the Reyleigh price model. By modifying the basic Reyleigh model, we can more accurately illustrate the economic phenomena with price varying. First, we research the dynamics of the modified Reyleigh model with time delay. By employing the normal form theory and center manifold theory, we obtain some testable results on these issues. The conclusion confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. Finally, some numerical simulations are given to illustrate the effectiveness of our results.

ON HOPF BIFURCATION OF A DELAYED PREDATOR–PREY SYSTEM WITH DIFFUSION

2013 ◽  
Vol 23 (02) ◽  
pp. 1350023 ◽  
Author(s):  
JIANXIN LIU ◽  
JUNJIE WEI
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Hopf Bifurcations ◽  
Center Manifold Theory ◽  
Predator Prey ◽  
Prey System ◽  
Normal Form Method ◽  
Manifold Theory

A delayed predator–prey system with diffusion and Dirichlet boundary conditions is considered. By regarding the growth rate a of prey as a main bifurcation parameter, we show that Hopf bifurcation occurs when the parameter a is varied. Then, by using the center manifold theory and normal form method, an explicit algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcating periodic solutions is derived.

Bifurcation Analysis for Neural Networks in Neutral Form

2016 ◽  
Vol 26 (06) ◽  
pp. 1650100 ◽  
Author(s):  
Hong-Bing Chen ◽  
Xiao-Ke Sun
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Center Manifold Theory ◽  
Neutral Form ◽  
Theoretical Predictions ◽  
Normal Form Method ◽  
The Stability ◽  
Manifold Theory

In this paper, a system of neural networks in neutral form with time delay is investigated. Further, by introducing delay [Formula: see text] as a bifurcation parameter, it is found that Hopf bifurcation occurs when [Formula: see text] is across some critical values. The direction of the Hopf bifurcations and the stability are determined by using normal form method and center manifold theory. Next, the global existence of periodic solution is established by using a global Hopf bifurcation result. Finally, an example is given to support the theoretical predictions.

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