BIFURCATION AND CONTROL IN A DIFFERENTIAL-ALGEBRAIC HARVESTED PREY-PREDATOR MODEL WITH STAGE STRUCTURE FOR PREDATOR

2008 ◽  
Vol 18 (10) ◽  
pp. 3159-3168 ◽  
Author(s):  
CHAO LIU ◽  
QINGLING ZHANG ◽  
YUE ZHANG ◽  
XIAODONG DUAN
Keyword(s):  
Stage Structure ◽  
Algebraic Model ◽  
Model System ◽  
Positive Equilibrium ◽  
Proposed Model ◽  
Stability And Instability ◽  
Differential Algebraic System ◽  
Predator Model ◽  
And Control ◽  
Predator System

A differential-algebraic model system which considers a prey-predator system with stage structure for a predator and a harvest effort on the mature predator is proposed. By using the differential-algebraic system and bifurcation theories, the local stability and instability mechanism of the proposed model system are investigated. With the purpose of stabilizing the proposed model system at the positive equilibrium, a state feedback controller is designed. Finally, a numerical simulation is carried out to show the consistency with theoretical analysis and illustrate the effectiveness of the proposed controller.

DYNAMICAL BEHAVIOR IN A HARVESTED DIFFERENTIAL-ALGEBRAIC PREY-PREDATOR MODEL

2009 ◽  
Vol 02 (04) ◽  
pp. 463-482 ◽  
Author(s):  
CHAO LIU ◽  
QINGLING ZHANG ◽  
XUE ZHANG ◽  
XIAODONG DUAN
Keyword(s):  
Bifurcation Theory ◽  
State Feedback ◽  
Dynamical Behavior ◽  
Algebraic Model ◽  
Economic Interest ◽  
Interior Equilibrium ◽  
Proposed Model ◽  
Differential Algebraic System ◽  
Predator Model ◽  
Predator System

A differential-algebraic model which considers a prey-predator system with harvest effort on predator is proposed. By using the differential-algebraic system theory and bifurcation theory, local stability of the proposed model around the interior equilibrium is investigated. Furthermore, the instability mechanism of the proposed model due to the variation of economic interest of harvesting is studied. With the purpose of stabilizing the proposed model around the interior equilibrium and maintaining the economic interest of harvesting at an ideal level, a state feedback controller is designed. Finally, numerical simulations are carried out to show the consistency with theoretical analysis.

scholarly journals Stability Analysis of a Harvested Prey-Predator Model with Stage Structure and Maturation Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Chao Liu ◽  
Qingling Zhang ◽  
James Huang
Keyword(s):  
Stability Analysis ◽  
Stage Structure ◽  
Model System ◽  
Positive Equilibrium ◽  
Predator Population ◽  
Maturation Delay ◽  
Effort Level ◽  
Selective Harvest ◽  
Proposed Model ◽  
Predator Model

A harvested prey-predator model with density-dependent maturation delay and stage structure for prey is proposed, where selective harvest effort on predator population is considered. Conditions which influence positiveness and boundedness of solutions of model system are analytically investigated. Criteria for existence of all equilibria and uniqueness of positive equilibrium are also studied. In order to discuss effects of maturation delay and harvesting on model dynamics, local stability analysis around all equilibria of the proposed model system is discussed due to variation of maturation delay and harvest effort level. Furthermore, global stability of positive equilibrium is investigated by utilizing an iterative technique. Finally, numerical simulations are carried out to show consistency with theoretical analysis.

scholarly journals The Dynamical Analysis of a Prey-Predator Model with a Refuge-Stage Structure Prey Population

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Raid Kamel Naji ◽  
Salam Jasim Majeed
Keyword(s):  
Mathematical Model ◽  
Equilibrium Points ◽  
Stage Structure ◽  
Dynamical Analysis ◽  
Local Bifurcation ◽  
Structure Population ◽  
Global Stability Analysis ◽  
Proposed Model ◽  
Predator Model ◽  
Predator System

We proposed and analyzed a mathematical model dealing with two species of prey-predator system. It is assumed that the prey is a stage structure population consisting of two compartments known as immature prey and mature prey. It has a refuge capability as a defensive property against the predation. The existence, uniqueness, and boundedness of the solution of the proposed model are discussed. All the feasible equilibrium points are determined. The local and global stability analysis of them are investigated. The occurrence of local bifurcation (such as saddle node, transcritical, and pitchfork) near each of the equilibrium points is studied. Finally, numerical simulations are given to support the analytic results.

scholarly journals Bifurcation and Feedback Control of an Exploited Prey-Predator System

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Uttam Das
Keyword(s):  
Theoretical Analysis ◽  
State Feedback ◽  
Dynamical Behavior ◽  
Algebraic Model ◽  
Positive Equilibrium ◽  
Economic Interest ◽  
Economic Profit ◽  
Proposed Model ◽  
Singularity Induced Bifurcation ◽  
Predator System

This paper makes an attempt to highlight a differential algebraic model in order to investigate the dynamical behavior of a prey-predator system due to the variation of economic interest of harvesting. In this regard, it is observed that the model exhibits a singularity induced bifurcation when economic profit is zero. For the purpose of stabilizing the proposed model at the positive equilibrium, a state feedback controller is therefore designed. Finally, some numerical simulations are carried out to show the consistency with theoretical analysis and to illustrate the effectiveness of the proposed controller.

scholarly journals An Impulsively Controlled Three-Species Prey-Predator Model with Stage Structure and Birth Pulse for Predator

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Yanyan Hu ◽  
Mei Yan ◽  
Zhongyi Xiang
Keyword(s):  
Small Amplitude ◽  
Floquet Theory ◽  
Stage Structure ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Birth Pulse ◽  
Impulsive Equation ◽  
Predator Model ◽  
Predator System

We investigate the dynamic behaviors of a two-prey one-predator system with stage structure and birth pulse for predator. By using the Floquet theory of linear periodic impulsive equation and small amplitude perturbation method, we show that there exists a globally asymptotically stable two-prey eradication periodic solution when the impulsive period is less than some critical value. Further, we study the permanence of the investigated model. Our results provide valuable strategy for biological economics management. Numerical analysis is also inserted to illustrate the results.

Hopf Bifurcation Analysis of a Three-Stage-Structured Prey-Predator System with Multi-Delays

2013 ◽  
Vol 756-759 ◽  
pp. 2857-2862
Author(s):  
Shun Yi Li ◽  
Wen Wu Liu
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Local Stability ◽  
Positive Equilibrium ◽  
Characteristic Equations ◽  
Numerical Examples ◽  
Predator Model ◽  
Predator System

A three-stage-structured prey-predator model with multi-delays is considered. The characteristic equations and local stability of the equilibrium are analyzed, and the conditions for the positive equilibrium occurring Hopf bifurcation are obtained by applying the theorem of Hopf bifurcation. Finally, numerical examples and brief conclusion are given.

Stationary Patterns of a Predator–Prey Model with Prey-Stage Structure and Prey-Taxis

2021 ◽  
Vol 31 (03) ◽  
pp. 2150038
Author(s):  
Meijun Chen ◽  
Huaihuo Cao ◽  
Shengmao Fu
Keyword(s):  
Stage Structure ◽  
Threshold Value ◽  
Turing Instability ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Self Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Stability And Instability ◽  
Negative Constant

In this paper, a predator–prey model with prey-stage structure and prey-taxis is proposed and studied. Firstly, the local stability of non-negative constant equilibria is analyzed. It is shown that non-negative equilibria have the same stability between ODE system and self-diffusion system, and self-diffusion does not have a destabilization effect. We find that there exists a threshold value [Formula: see text] such that the positive equilibrium point of the model becomes unstable when the prey-taxis rate [Formula: see text], hence the taxis-driven Turing instability occurs. Furthermore, by applying Crandall–Rabinowitz bifurcation theory, the existence, the stability and instability, and the turning direction of bifurcating steady state are investigated in detail. Finally, numerical simulations are provided to support the mathematical analysis.

scholarly journals Permanence and Stability of an Age-Structured Prey-Predator System with Delays

2007 ◽  
Vol 2007 ◽  
pp. 1-15 ◽  
Author(s):  
Liming Cai ◽  
Xuezhi Li ◽  
Xinyu Song ◽  
Jingyuan Yu
Keyword(s):  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Persistence Theory ◽  
Age Structured ◽  
Mathematical Analyses ◽  
Predator Model ◽  
Model Equations ◽  
Predator System

An age-structured prey-predator model with delays is proposed and analyzed. Mathematical analyses of the model equations with regard to boundedness of solutions, permanence, and stability are analyzed. By using the persistence theory for infinite-dimensional systems, the sufficient conditions for the permanence of the system are obtained. By constructing suitable Lyapunov functions and using an iterative technique, sufficient conditions are also obtained for the global asymptotic stability of the positive equilibrium of the model.

scholarly journals Hopf bifurcation analysis in a stage-structure predator–prey model with two time delay

2022 ◽  
Vol 355 ◽  
pp. 03048
Author(s):  
Bochen Han ◽  
Shengming Yang ◽  
Guangping Zeng
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Time Delays ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Predator Model

In this paper, we consider a predator-prey system with two time delays, which describes a prey–predator model with parental care for predators. The local stability of the positive equilibrium is analysed. By choosing the two time delays as the bifurcation parameter, the existence of Hopf bifurcation is studied. Numerical simulations show the positive equilibrium loses its stability via the Hopf bifurcation when the time delay increases beyond a threshold.

scholarly journals The Modelling and Control of a Singular Biological Economic System in a Polluted Environment

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Yi Zhang ◽  
Yueming Jie ◽  
Xinyou Meng
Keyword(s):  
Bifurcation Theory ◽  
State Feedback ◽  
Singular Systems ◽  
Stage Structure ◽  
Fishing Effort ◽  
Positive Equilibrium ◽  
Economic Profit ◽  
Singularity Induced Bifurcation ◽  
And Control ◽  
Theoretical Results

A singular biological economic model with harvesting and stage structure is presented. The local stability of equilibriums of the system is investigated when the economic profit parameter is zero, and the conditions of the singularity induced bifurcation occurring at the positive equilibrium are obtained by the singular systems theory and bifurcation theory. In order to eliminate the singularity induced bifurcation, a state feedback controller is designed by controlling the fishing effort. At last, an application example is given to illustrate the validity of the theoretical results.

Sign in / Sign up

Export Citation Format

Share Document