Optimal harvesting of a prey–predator model with variable carrying capacity

2017 ◽  
Vol 10 (05) ◽  
pp. 1750069 ◽  
Author(s):  
Chaity Ganguli ◽  
T. K. Kar ◽  
P. K. Mondal
Keyword(s):  
Carrying Capacity ◽  
Intraspecific Competition ◽  
Dynamical Behavior ◽  
Optimal Harvesting ◽  
Main Concern ◽  
Interior Equilibrium ◽  
Optimal Harvest ◽  
Common Resource ◽  
Predator Model ◽  
Optimal Harvest Policy

This work deals with a prey–predator model in an environment where the carrying capacities are assumed to be variable with time and one species feeds upon the other. Independent harvesting efforts are applied in either species and asymmetrical intraguild predation occurs. A common resource is consumed by two competing species and at the same time predator also consumes the prey. At first we discuss the model under constant carrying capacity and make the conclusion that no limit cycle exists in this case. Then we discuss the model without intraspecific competition. Our main concern is to cover the above mentioned two cases together, i.e. the model with variable carrying capacity and intraspecific competition. We determine the steady states and examine the dynamical behavior. We also analyze the local and global stability of the interior equilibrium by Routh–Hurwitz criterion and a suitable Lyapunov function respectively. A Hopf bifurcation occurs with respect to a parameter which is the ratio of predator’s and prey’s intrinsic growth rate. The possibility of bionomic equilibrium has been considered. The optimal harvest policy is formulated and solved with Pontryagin’s maximum principle. Some numerical simulations are given to explain most of the analytical results.

scholarly journals Dynamics Analysis and Optimality in Selective Harvesting Predator-Prey Model With Modified Leslie-Gower and Holling-Type II

2019 ◽  
Vol 6 (1) ◽  
pp. 1-17
Author(s):  
W. Abid ◽  
R. Yafia ◽  
M. A. Aziz-Alaoui ◽  
Ahmed Aghriche
Keyword(s):  
Existence Of Solutions ◽  
Optimal Harvesting ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Optimal Harvest ◽  
Predator Model ◽  
Maximal Principle ◽  
Theoretical Results ◽  
Optimal Harvest Policy

AbstractIn this work, we consider the optimal harvesting and stability problems of a prey-predator model with modified Leslie-Gower and Holling-type II functional response. The model is governed by a system of three differential equations which describe the interactions between prey, predator and harvesting effort. Boundedness and existence of solutions for this system are showed. The existence and local stability of the possible steady states are analyzed and the conditions of global stability of the interior equilibrium are established by using the Lyapunov function, we prove also the occurrence of Hopf bifurcation at this point. By using the Pontryagin’s maximal principle, we formulate and we solve the problem of the optimal harvest policy. In the end, some numerical simulations are given to support our theoretical results.

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 Stability and Selective Harvesting of a Phytoplankton-Zooplankton System

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Yong Wang ◽  
Hongbin Wang
Keyword(s):  
Hopf Bifurcation ◽  
Carrying Capacity ◽  
Optimal Harvesting ◽  
Control Constraints ◽  
State Equations ◽  
Optimal Harvest ◽  
Transcritical Bifurcations ◽  
Bodies Of Water ◽  
The Impact ◽  
Optimal Harvest Policy

Considering that some zooplankton can be harvested for food in some bodies of water, a phytoplankton-zooplankton model with continuous harvesting of zooplankton only is proposed and investigated. By using environmental carrying capacity as a parameter, possible dynamic behaviors, such as stability, global stability, Hopf bifurcation, and transcritical bifurcations, are analyzed. The optimal harvesting policy is disposed by imposing a tax per unit biomass of zooplankton. The problem of determining the optimal harvest policy is solved by using Pontryagin's maximum principle subject to the state equations and the control constraints, and the impact of tax is also discussed. Finally, some numerical simulations are performed to justify analytical findings.

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 Analysis of a Fishery Model with two competing prey species in the presence of a predator species for Optimal Harvesting

2018 ◽  
Vol 31 ◽  
pp. 08008 ◽  
Author(s):  
Sutimin ◽  
Siti Khabibah ◽  
Dita Anis Munawwaroh
Keyword(s):  
Ecological Model ◽  
Dynamical Behavior ◽  
Fish Population ◽  
Optimal Harvesting ◽  
Globally Asymptotically Stable ◽  
Predator Species ◽  
Optimal Harvest ◽  
The Stability ◽  
Fishery Model ◽  
Maximal Principle

A harvesting fishery model is proposed to analyze the effects of the presence of red devil fish population, as a predator in an ecosystem. In this paper, we consider an ecological model of three species by taking into account two competing species and presence of a predator (red devil), the third species, which incorporates the harvesting efforts of each fish species. The stability of the dynamical system is discussed and the existence of biological and bionomic equilibrium is examined. The optimal harvest policy is studied and the solution is derived in the equilibrium case applying Pontryagin’s maximal principle. The simulation results is presented to simulate the dynamical behavior of the model and show that the optimal equilibrium solution is globally asymptotically stable. The results show that the optimal harvesting effort is obtained regarding to bionomic and biological equilibrium.

scholarly journals Dynamical Behavior and Stability Analysis in a Stage-Structured Prey Predator Model with Discrete Delay and Distributed Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Chao Liu ◽  
Qingling Zhang
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Local Stability ◽  
Dynamical Behavior ◽  
Distributed Delay ◽  
Stable Limit ◽  
Discrete Delay ◽  
Normal Form Theory ◽  
Interior Equilibrium ◽  
Predator Model

We propose a prey predator model with stage structure for prey. A discrete delay and a distributed delay for predator described by an integral with a strong delay kernel are also considered. Existence of two feasible boundary equilibria and a unique interior equilibrium are analytically investigated. By analyzing associated characteristic equation, local stability analysis of boundary equilibrium and interior equilibrium is discussed, respectively. It reveals that interior equilibrium is locally stable when discrete delay is less than a critical value. According to Hopf bifurcation theorem for functional differential equations, it can be found that model undergoes Hopf bifurcation around the interior equilibrium when local stability switch occurs and corresponding stable limit cycle is observed. Furthermore, directions of Hopf bifurcation and stability of the bifurcating periodic solutions are studied based on normal form theory and center manifold theorem. Numerical simulations are carried out to show consistency with theoretical analysis.

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.

DYNAMIC ANALYSIS IN A HARVESTED DIFFERENTIAL-ALGEBRAIC PREY–PREDATOR MODEL

2009 ◽  
Vol 09 (01) ◽  
pp. 123-140 ◽  
Author(s):  
CHAO LIU ◽  
QINGLING ZHANG ◽  
XUE ZHANG
Keyword(s):  
Theoretical Analysis ◽  
Local Stability ◽  
Dynamical Behavior ◽  
Algebraic Model ◽  
Economic Interest ◽  
Instability Mechanism ◽  
Biological Resource ◽  
Interior Equilibrium ◽  
Proposed Model ◽  
Predator Model

Nowadays, the biological resource in the prey–predator ecosystem is commercially harvested and sold with the aim of achieving economic interest. Furthermore, the harvest effort is usually influenced by the variation of economic interest of harvesting. In this paper, a differential–algebraic model is proposed, which is utilized to investigate the dynamical behavior of the prey–predator ecosystem due to the variation of economic interest of harvesting. By discussing the local stability of the proposed model around the interior equilibrium, the instability mechanism of harvested prey–predator ecosystem 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 feedback controller is designed. Finally, numerical simulations are carried out to demonstrate consistency with the theoretical analysis.

scholarly journals Harvesting on Facultative Mutualist Prey Species in Presence of a Predator

2017 ◽  
Vol 1 (1) ◽  
pp. 1
Author(s):  
Saroj Kumar Chattopadhyay
Keyword(s):  
Maximum Principle ◽  
Numerical Simulations ◽  
Prey Species ◽  
Pontryagin’S Maximum Principle ◽  
Pontryagin's Maximum Principle ◽  
Interior Equilibrium ◽  
Optimal Harvest ◽  
The Stability ◽  
Optimal Harvest Policy

<p><em>This paper proposes a model with two preys of facultative mutualist type and one predator. Linear predation functions are considered and preys are only considered to be harvested. The stability of the model is analyzed theoretically and numerically in this paper. The optimal harvest policy is studied and the solution is derived in the interior equilibrium case using Pontryagin’s maximum principle. Finally, some numerical simulations are discussed.</em></p>

scholarly journals Fear effect in discrete prey-predator model incorporating square root functional response

2021 ◽  
Vol 2 (2) ◽  
pp. 51-57
Author(s):  
P.K. Santra
Keyword(s):  
Functional Response ◽  
Stability Condition ◽  
Discrete Model ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Square Root ◽  
Bifurcation Diagrams ◽  
Fear Effect ◽  
Hypothetical Data ◽  
Predator Model

In this work, an interaction between prey and its predator involving the effect of fear in presence of the predator and the square root functional response is investigated. Fixed points and their stability condition are calculated. The conditions for the occurrence of some phenomena namely Neimark-Sacker, Flip, and Fold bifurcations are given. Base on some hypothetical data, the numerical simulations consist of phase portraits and bifurcation diagrams are demonstrated to picturise the dynamical behavior. It is also shown numerically that rich dynamics are obtained by the discrete model as the effect of fear.

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