Analysis of Prey-Predator Three Species Fishery Model with Harvesting Including Prey Refuge and Migration

2016 ◽  
Vol 26 (02) ◽  
pp. 1650022 ◽  
Author(s):  
Sankar Kumar Roy ◽  
Banani Roy
Keyword(s):  
Fishing Effort ◽  
Predator Population ◽  
Prey Refuge ◽  
Resource Rent ◽  
Interior Equilibrium ◽  
Dynamic Framework ◽  
Resource Sustainability ◽  
And Migration ◽  
Fishery Model ◽  
Type Ii Functional Response

In this article, a prey-predator system with Holling type II functional response for the predator population including prey refuge region has been analyzed. Also a harvesting effort has been considered for the predator population. The density-dependent mortality rate for the prey, predator and super predator has been considered. The equilibria of the proposed system have been determined. Local and global stabilities for the system have been discussed. We have used the analytic approach to derive the global asymptotic stabilities of the system. The maximal predator per capita consumption rate has been considered as a bifurcation parameter to evaluate Hopf bifurcation in the neighborhood of interior equilibrium point. Also, we have used fishing effort to harvest predator population of the system as a control to develop a dynamic framework to investigate the optimal utilization of the resource, sustainability properties of the stock and the resource rent is earned from the resource. Finally, we have presented some numerical simulations to verify the analytic results and the system has been analyzed through graphical illustrations.

ECOLOGICAL SUSTAINABILITY OF AN OPTIMAL CONTROLLED SYSTEM INCORPORATING PARTIAL CLOSURE FOR THE POPULATIONS

2015 ◽  
Vol 23 (03) ◽  
pp. 355-384 ◽  
Author(s):  
KUNAL CHAKRABORTY ◽  
SAMADYUTI HALDAR ◽  
T. K. KAR
Keyword(s):  
Fishing Effort ◽  
Predator Population ◽  
Partial Closure ◽  
Interior Equilibrium ◽  
Dynamic Framework ◽  
Bifurcation Phenomenon ◽  
Control Scheme ◽  
Order Scheme ◽  
Optimal Control Scheme ◽  
Fishery Model

This paper describes a prey–predator fishery model incorporating partial closure for the populations. It is assumed that the predator population partially dependent on a logistically growing resource with Beddington–De Angelis type functional response. The proposed system also reflects the dynamic interaction between the net economic revenue and the fishing effort used to harvest the populations. The steady states of the system are determined and the dynamic behavior of the system is discussed. The existence of Hopf bifurcation phenomenon is examined at the interior equilibrium point of the proposed system. We have adopted partial closure for the populations as a controlling instrument to regulate the harvesting of the populations. A dynamic framework towards the optimal utilization of the resource is developed using Pontryagin's maximum principle. The optimal system is numerically solved using an iterative method with Runge–Kutta fourth-order scheme. Simulation results show that the optimal control scheme can achieve a sustainable ecosystem. Results are analyzed with the help of graphical illustrations.

scholarly journals Prey-predator model in drainage system with migration and harvesting

2021 ◽  
Vol 8 (1) ◽  
pp. 152-167
Author(s):  
Banani Roy ◽  
Sankar Kumar Roy
Keyword(s):  
Functional Response ◽  
Control Parameter ◽  
Drainage System ◽  
Fishing Effort ◽  
Future Research ◽  
Generalist Predator ◽  
Predator Population ◽  
Resource Rent ◽  
Dynamic Framework ◽  
Predator Model

Abstract In this paper, we consider a prey-predator model with a reserve region of predator where generalist predator cannot enter. Based on the intake capacity of food and other factors, we introduce the predator population which consumes the prey population with Holling type-II functional response; and generalist predator population consumes the predator population with Beddington-DeAngelis functional response. The density-dependent mortality rate for prey and generalist predator are considered. The equilibria of proposed system are determined. Local stability for the system are discussed. The environmental carrying capacity is considered as a bifurcation parameter to evaluate Hopf bifurcation in the neighbourhood at an interior equilibrium point. Here the fishing effort is used as a control parameter to harvest the generalist predator population of the system. With the help of this control parameter, a dynamic framework is developed to investigate the optimal utilization of resources, sustainability properties of the stock and the resource rent. Finally, we present a numerical simulation to verify the analytical results, and the system is analyzed through graphical illustrations. The main findings with future research directions are described at last.

REGULATION OF A PREY–PREDATOR FISHERY INCORPORATING PREY REFUGE BY TAXATION: A DYNAMIC REACTION MODEL

2011 ◽  
Vol 19 (03) ◽  
pp. 417-445 ◽  
Author(s):  
KUNAL CHAKRABORTY ◽  
MILON CHAKRABORTY ◽  
T. K. KAR
Keyword(s):  
Optimal Taxation ◽  
Social Benefit ◽  
Reaction Model ◽  
Fishing Effort ◽  
Model System ◽  
Prey Refuge ◽  
Predator Species ◽  
Taxation Policy ◽  
Proposed Model ◽  
Fishery Model

This paper, describes a prey–predator fishery model incorporating prey refuge. The proposed model reflecting the dynamic interaction between the net economic revenue and the fishing effort used to harvest the prey species in the presence of predation and a suitable tax. The steady states of the system are determined and the dynamic behavior of the model system is discussed. The occurrence of Hopf bifurcation of the proposed model system is examined through considering density-dependent mortality for the predator as bifurcation parameter. The optimal taxation policy is formulated and solved with the help of Pontryagin's maximal principle. The objective of the paper is to maximize the monetary social benefit as well as prevent the predator species from extinction, keeping the ecological balance. Results are illustrated with the help of numerical examples.

scholarly journals Global Dynamics of an Exploited Prey-Predator Model with Constant Prey Refuge

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Uttam Das ◽  
T. K. Kar ◽  
U. K. Pahari
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Global Dynamics ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Species ◽  
Interior Equilibrium ◽  
Predator Model ◽  
Theoretical Results ◽  
Type Ii Functional Response

This paper describes a prey-predator model with Holling type II functional response incorporating constant prey refuge and harvesting to both prey and predator species. We have analyzed the boundedness of the system and existence of all possible feasible equilibria and discussed local as well as global stabilities at interior equilibrium of the system. The occurrence of Hopf bifurcation of the system is examined, and it was observed that the bifurcation is either supercritical or subcritical. Influences of prey refuge and harvesting efforts are also discussed. Some numerical simulations are carried out for the validity of theoretical results.

DYNAMIC CONSEQUENCES OF PREY REFUGIA IN A TWO-PREDATOR–ONE-PREY SYSTEM

2013 ◽  
Vol 21 (02) ◽  
pp. 1350013 ◽  
Author(s):  
T. K. KAR ◽  
ABHIJIT GHORAI ◽  
SOOVOOJEET JANA
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Prey Refuge ◽  
Center Manifold Theorem ◽  
Interior Equilibrium ◽  
Prey System ◽  
The Stability ◽  
Normal Form Methods ◽  
Type Ii Functional Response

We consider a two predator and one prey model with Holling type II functional response incorporating a constant prey refuge. Depending upon the constant prey refuge m, which provides a criterion for protecting m of prey from predation, sufficient conditions for stability and global stability of equilibria are obtained. We find the critical value of this refuge parameter m for which the dynamical system undergoes a Hopf bifurcation and then makes use of center manifold theorem and normal form methods to find the direction of the Hopf bifurcation as well as the stability of the resulting limit cycle. The influence of the prey refuge parameter is also investigated at the interior equilibrium. Numerical simulations were carried out to illustrate and support the analytical results.

scholarly journals Dynamical Behaviour in Two Prey-Predator System with Persistence

2016 ◽  
Vol 16 ◽  
pp. 20-37
Author(s):  
V. Madhusudanan ◽  
S. Vijaya
Keyword(s):  
Limit Cycle ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Behaviour ◽  
Positive Equilibrium ◽  
Predator Population ◽  
Interior Equilibrium ◽  
Trivial Equilibrium ◽  
Necessary And Sufficient

In this work, the dynamical behavior of the system with two preys and one predator population is investigated. The predator exhibits a Holling type II response to one prey which is harvested and a Beddington-DeAngelis functional response to the other prey. The boundedness of the system is analyzed. We examine the occurrence of positive equilibrium points and stability of the system at those points. At trivial equilibrium E0and axial equilibrium (E1); the system is found to be unstable. Also we obtain the necessary and sufficient conditions for existence of interior equilibrium point (E6) and local and global stability of the system at the interior equilibrium (E6): Depending upon the existence of limit cycle, the persistence condition is established for the system. The numerical simulation infer that varying the parameters such as e and λ1it is possible to change the dynamical behavior of the system from limit cycle to stable spiral. It is also observed that the harvesting rate plays a crucial role in stabilizing the system.

Modeling the Effect of Fear in a Prey–Predator System with Prey Refuge and Gestation Delay

2019 ◽  
Vol 29 (14) ◽  
pp. 1950195 ◽  
Author(s):  
Ankit Kumar ◽  
Balram Dubey
Keyword(s):  
Hopf Bifurcation ◽  
Prey Population ◽  
Reproduction Rate ◽  
Predator Population ◽  
Maximum Lyapunov Exponent ◽  
Stable Limit ◽  
Prey Refuge ◽  
Gestation Delay ◽  
Fear Effect ◽  
Predator System

Recently, some field experiments and studies show that predators affect prey not only by direct killing, they induce fear in prey which reduces the reproduction rate of prey species. Considering this fact, we propose a mathematical model to study the fear effect and prey refuge in prey–predator system with gestation time delay. It is assumed that prey population grows logistically in the absence of predators and the interaction between prey and predator is followed by Crowley–Martin type functional response. We obtained the equilibrium points and studied the local and global asymptotic behaviors of nondelayed system around them. It is observed from our analysis that the fear effect in the prey induces Hopf-bifurcation in the system. It is concluded that the refuge of prey population under a threshold level is lucrative for both the species. Further, we incorporate gestation delay of the predator population in the model. Local and global asymptotic stabilities for delayed model are carried out. The existence of stable limit cycle via Hopf-bifurcation with respect to delay parameter is established. Chaotic oscillations are also observed and confirmed by drawing the bifurcation diagram and evaluating maximum Lyapunov exponent for large values of delay parameter.

A Predator-Prey Model Incorporating Prey Refuge and Allee Effect

2015 ◽  
Vol 713-715 ◽  
pp. 1534-1539 ◽  
Author(s):  
Rui Ning Fan
Keyword(s):  
Time Delay ◽  
Time Lag ◽  
Functional Responses ◽  
Prey Refuge ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
Discrete Time Delay ◽  
Prey Interaction

The effect of refuge used by prey has a stabilizing impact on population dynamics and the effect of time delay has its destabilizing influences. Little attention has been paid to the combined effects of prey refuge and time delay on the dynamic consequences of the predator-prey interaction. Here, a predator-prey model with a class of functional responses was studied by using the analytical approach. The refuge is considered as protecting a constant proportion of prey and the discrete time delay is the gestation period. We evaluated both effects with regard to the local stability of the interior equilibrium point of the considered model. The results showed that the effect of prey refuge has stronger influences than that of time delay on the considered model when the time lag is smaller than the threshold. However, if the time lag is larger than the threshold, the effect of time delay has stronger influences than that of refuge used by prey.

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