scholarly journals Stability and optimal harvesting of a prey-predator model with stage structure for predator

2005 ◽  
Vol 32 (3) ◽  
pp. 279-291 ◽  
Author(s):  
Tapan Kumar Kar
Keyword(s):  
Stage Structure ◽  
Optimal Harvesting ◽  
Predator Model

scholarly journals Dynamics and Optimal Taxation Control in a Bioeconomic Model with Stage Structure and Gestation Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Yi Zhang ◽  
Qingling Zhang ◽  
Fenglan Bai
Keyword(s):  
Optimal Taxation ◽  
Stable Limit Cycle ◽  
Stage Structure ◽  
Model System ◽  
Optimal Harvesting ◽  
Stable Limit ◽  
Gestation Delay ◽  
Interior Equilibrium ◽  
Control Instrument ◽  
Predator Model

A prey-predator model with gestation delay, stage structure for predator, and selective harvesting effort on mature predator is proposed, where taxation is considered as a control instrument to protect the population resource in prey-predator biosystem from overexploitation. It shows that interior equilibrium is locally asymptotically stable when the gestation delay is zero, and there is no periodic orbit within the interior of the first quadrant of state space around the interior equilibrium. An optimal harvesting policy can be obtained by virtue of Pontryagin's Maximum Principle without considering gestation delay; on the other hand, the interior equilibrium of model system loses as gestation delay increases through critical certain threshold, a phenomenon of Hopf bifurcation occurs, and a stable limit cycle corresponding to the periodic solution of model system is also observed. Finally, numerical simulations are carried out to show consistency with theoretical analysis.

DYNAMICAL BEHAVIOR OF A HARVESTED PREY-PREDATOR MODEL WITH STAGE STRUCTURE AND DISCRETE TIME DELAY

2009 ◽  
Vol 17 (04) ◽  
pp. 759-777 ◽  
Author(s):  
CHAO LIU ◽  
QINGLING ZHANG ◽  
JAMES HUANG ◽  
WANSHENG TANG
Keyword(s):  
Time Delay ◽  
Discrete Time ◽  
Dynamical Behavior ◽  
Stage Structure ◽  
Model System ◽  
Optimal Harvesting ◽  
Gestation Delay ◽  
Interior Equilibrium ◽  
Discrete Time Delay ◽  
Predator Model

A prey-predator model with stage structure for prey and selective harvest effort on predator is proposed, in which gestation delay is considered and taxation is used as a control instrument to protect the population from overexploitation. It is established that when the discrete time delay is zero, the model system is stable around the interior equilibrium and an optimal harvesting policy is discussed with the help of Pontryagin's maximum principle; On the other hand, stability switch of the model system due to the variation of discrete time delay is also studied, which reveals that the discrete time delay has a destabilizing effect. As the discrete time delay increases through a certain threshold, a phenomenon of Hopf bifurcation occurs and a limit cycle corresponding to the periodic solution of model system is also observed. Numerical simulations are carried out to show the consistency with theoretical analysis.

scholarly journals Optimal harvesting policy of a prey–predator model with Crowley–Martin-type functional response and stage structure in the predator

2018 ◽  
Vol 23 (4) ◽  
pp. 493-514 ◽  
Author(s):  
Balram Dubey ◽  
Shikhar Agarwal ◽  
Ankit Kumar
Keyword(s):  
Three Dimensional ◽  
Stage Structure ◽  
Optimal Level ◽  
Effective Control ◽  
Optimal Harvesting ◽  
Catch Per Unit Effort ◽  
Optimal Harvesting Policy ◽  
Control Instrument ◽  
Long Time ◽  
Predator Model

In this paper, a three-dimensional dynamical model consisting of a prey, a mature predator, and an immature predator is proposed and analysed. The interaction between prey and mature predator is assumed to be of the Crowley–Martin type, and both the prey and mature predator are harvested according to catch-per-unit-effort (CPUE) hypothesis. Steady state of the system is obtained, stability analysis (local and global both) are discussed to explore the long-time behaviour of the system. The optimal harvesting policy is also discussed with the help of Pontryagin's maximum principle. The harvesting effort is taken as an effective control instrument to preserve prey and predator and to maintain them at an optimal level.

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.

Prey–predator model for optimal harvesting with functional response incorporating prey refuge

2015 ◽  
Vol 09 (01) ◽  
pp. 1650014 ◽  
Author(s):  
G. S. Mahapatra ◽  
P. Santra
Keyword(s):  
Functional Response ◽  
System Parameter ◽  
Optimal Harvesting ◽  
Functional Responses ◽  
Prey Refuge ◽  
Numerical Examples ◽  
Pictorial Presentation ◽  
Constant Form ◽  
Predator Model ◽  
Predator System

This paper presents a prey–predator model considering the predator interacting with non-refuges prey by class of functional responses. Here we also consider harvesting for only non-refuges prey. We discuss the equilibria of the model, and their stability for hiding prey either in constant form or proportional to the densities of prey population. We also investigate various possibilities of bionomic equilibrium and optimal harvesting policy. Finally we present numerical examples with pictorial presentation of the various effects of the prey–predator system parameter.

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