HOPF BIFURCATION AND CHAOS OF A HYBRID THREE-SPECIES FOOD CHAIN WITH TIME DELAYED INTERVENTION

2012 ◽  
Vol 22 (05) ◽  
pp. 1250122
Author(s):  
YUNXIAN DAI ◽  
YIPING LIN
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Food Chain ◽  
Positive Equilibrium ◽  
Bifurcation And Chaos ◽  
Analytical Results ◽  
Ratio Dependent ◽  
Delayed Intervention

This paper is concerned with the effect of time delayed intervention on a hybrid ratio dependent three-species food chain. We obtain the conditions under which the positive equilibrium is stable and Hopf bifurcation occurs with the change of delay parameter. Hassard's method is used to obtain the direction and stability of the Hopf bifurcation. Finally, numerical simulations are carried out to support the analytical results and chaotic behaviors are observed.

Rank-One Chaos in a Periodically Kicked Three-Species Food Chain with Time-Delay

2020 ◽  
Vol 30 (03) ◽  
pp. 2050038
Author(s):  
Ping Yang ◽  
Juan Fang ◽  
Yunxian Dai ◽  
Yiping Lin
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Food Chain ◽  
Strange Attractor ◽  
Positive Equilibrium ◽  
Rank One ◽  
Delayed System

This paper is devoted to studying the problem of rank-one strange attractor in a three-species food chain with time-delay. The conditions for the existence of positive equilibrium and Hopf bifurcation are presented. By using the theory of rank-one maps formulated by Wang and Young in 2001, and then developed by us to the time-delayed system, the conditions for the system having rank-one strange attractor are obtained under periodically kicked system. Numerical simulations are presented to demonstrate the analytic results.

Hopf Bifurcation Analysis and Stability for a Ratio-Dependent Predator–Prey Diffusive System with Time Delay

2020 ◽  
Vol 30 (03) ◽  
pp. 2050037
Author(s):  
Longyue Li ◽  
Yingying Mei ◽  
Jianzhi Cao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Periodic Solutions ◽  
Positive Equilibrium ◽  
Dimensional System ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Ratio Dependent ◽  
Low Dimensional

In this paper, we are focused on a new ratio-dependent predator–prey system that introduced the diffusive and time delay effect simultaneously. By analyzing the characteristic equations and the distribution of eigenvalues, we examine the stability and boundary of positive equilibrium states, and the existence of spatially homogeneous and spatially inhomogeneous bifurcating periodic solutions, respectively. Further, we prove that when [Formula: see text], the system has Hopf bifurcation at the positive equilibrium state. By using the center manifold reduction, we simplify the system so that we can convert an infinite-dimensional system into a low-dimensional finite-dimensional system. By using the normal form theory, we obtain explicit expressions for the direction, stability and period of Hopf bifurcation periodic solutions. Finally, we have illustrated the main results in this thesis by numerical examples, our work may provide some useful measures to save time or cost and to control the ecosystem.

scholarly journals Pattern Formation in a Diffusive Ratio-Dependent Holling-Tanner Predator-Prey Model with Smith Growth

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Bo Yang
Keyword(s):  
Boundary Condition ◽  
Hopf Bifurcation ◽  
Turing Instability ◽  
Turing Patterns ◽  
Positive Equilibrium ◽  
Flux Boundary Condition ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

The spatiotemporal dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey model with Smith growth subject to zero-flux boundary condition are investigated analytically and numerically. The asymptotic stability of the positive equilibrium and the existence of Hopf bifurcation around the positive equilibrium are shown; the conditions of Turing instability are obtained. And with the help of numerical simulations, it is found that the model exhibits complex pattern replication: stripes, spots-stripes mixtures, and spots Turing patterns.

A RATIO-DEPENDENT PREDATOR-PREY MODEL WITH DELAY AND HARVESTING

2010 ◽  
Vol 18 (02) ◽  
pp. 437-453 ◽  
Author(s):  
A. K. MISRA ◽  
B. DUBEY
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Functional Response ◽  
Bifurcation Parameter ◽  
Discrete Delay ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

In this paper a predator-prey model with discrete delay and harvesting of predator is proposed and analyzed by considering ratio-dependent functional response. Conditions of existence of various equilibria and their stability have been discussed. By taking delay as a bifurcation parameter, the system is found to undergo a Hopf bifurcation. Numerical simulations are also performed to illustrate the results.

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.

scholarly journals Stability and Global Hopf Bifurcation Analysis on a Ratio-Dependent Predator-Prey Model with Two Time Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-15
Author(s):  
Huitao Zhao ◽  
Yiping Lin ◽  
Yunxian Dai
Keyword(s):  
Hopf Bifurcation ◽  
Time Delays ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Global Hopf Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

A ratio-dependent predator-prey model with two time delays is studied. By means of an iteration technique, sufficient conditions are obtained for the global attractiveness of the positive equilibrium. By comparison arguments, the global stability of the semitrivial equilibrium is addressed. By using the theory of functional equation and Hopf bifurcation, the conditions on which positive equilibrium exists and the quality of Hopf bifurcation are given. Using a global Hopf bifurcation result of Wu (1998) for functional differential equations, the global existence of the periodic solutions is obtained. Finally, an example for numerical simulations is also included.

Turing Instability and Hopf Bifurcation for the General Brusselator System

2011 ◽  
Vol 255-260 ◽  
pp. 2126-2130
Author(s):  
Gai Hui Guo ◽  
Bing Fang Li
Keyword(s):  
Hopf Bifurcation ◽  
Boundary Conditions ◽  
Numerical Simulations ◽  
Equilibrium Solution ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Neumann Boundary Conditions ◽  
Homogeneous Equilibrium ◽  
Analytical Results

The Brusselator system subject to homogeneous Neumann boundary conditions is investigated. It is firstly shown that the homogeneous equilibrium solution becomes Turing unstable or diffusively unstable when parameters are chosen properly. Then the existence of Hopf bifurcation to the ODE and PDE models is obtained. Examples of numerical simulations are also shown to support and supplement the analytical results.

ECO-EPIDEMIOLOGICAL MODELS OF SALTON SEA WITH INFECTED PREY

2013 ◽  
Vol 21 (01) ◽  
pp. 1350003 ◽  
Author(s):  
Q. J. A. KHAN ◽  
E. BALAKRISHNAN ◽  
AZZA HAMOOD AL HARTHI
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Disease Transmission ◽  
Salton Sea ◽  
Bifurcation Parameter ◽  
Epidemiological Models ◽  
Analytical Results ◽  
Infected Prey

Two models for the interaction of susceptible and infected Tilapia population with Pelican population are studied. Here, we considered that Pelican interact with both susceptible and infected Tilapia in proportion to their abundance. Stability near nonzero equilibria is presented. In the second model, time delay is incorporated in the disease transmission term and Hopf bifurcation is analyzed by taking time delay as a bifurcation parameter. Numerical simulations are performed to support the analytical results.

scholarly journals Stability and Bifurcation in a Delayed Holling-Tanner Predator-Prey System with Ratio-Dependent Functional Response

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Juan Liu ◽  
Zizhen Zhang ◽  
Ming Fu
Keyword(s):  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Center Manifold ◽  
Predator Prey ◽  
Local Asymptotic Stability ◽  
Prey System ◽  
Ratio Dependent ◽  
The Stability

We analyze a delayed Holling-Tanner predator-prey system with ratio-dependent functional response. The local asymptotic stability and the existence of the Hopf bifurcation are investigated. Direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are studied by deriving the equation describing the flow on the center manifold. Finally, numerical simulations are presented for the support of our analytical findings.

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