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.

Parameter uncertainty in biomathematical model described by one-prey two-predator system with mutualism

2017 ◽  
Vol 10 (06) ◽  
pp. 1750082
Author(s):  
D. Pal ◽  
G. S. Mahapatra ◽  
G. P. Samanta
Keyword(s):  
Maximum Principle ◽  
Parameter Uncertainty ◽  
Equilibrium Points ◽  
Optimal Harvesting ◽  
Biological Parameters ◽  
Optimal Conditions ◽  
Stability Criteria ◽  
Numerical Examples ◽  
Predator Model ◽  
Predator System

In this paper, a three-species system consisting of two predators which are in mutualism with each other and preying on the same single prey is considered. Also, the prey and first predator are harvested under optimal conditions. The values of the biological parameters depend on the collection of data from the experts as well as on the nature of the environment in which prey–predator system are considered. So the biological parameters are not precise in reality. This paper presents a different approach to study the prey–predator model with imprecise biological parameters. All the possible equilibrium points are identified and the local as well as global stability criteria under impreciseness are discussed. The possibility of existence of bionomic equilibrium is discussed. The optimal harvesting policy is studied using Pontryagin’s maximum principle. Numerical examples are provided to support the proposed approach.

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.

scholarly journals Dynamical Analysis of a Class of Prey-Predator Model with Beddington-DeAngelis Functional Response, Stochastic Perturbation, and Impulsive Toxicant Input

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-18 ◽  
Author(s):  
Feifei Bian ◽  
Wencai Zhao ◽  
Yi Song ◽  
Rong Yue
Keyword(s):  
Periodic Solutions ◽  
Functional Response ◽  
Regime Switching ◽  
Sufficient Conditions ◽  
Stochastic Perturbation ◽  
Dynamical Analysis ◽  
Positive Periodic Solutions ◽  
Noise Perturbation ◽  
Predator Model ◽  
Predator System

A stochastic prey-predator system in a polluted environment with Beddington-DeAngelis functional response is proposed and analyzed. Firstly, for the system with white noise perturbation, by analyzing the limit system, the existence of boundary periodic solutions and positive periodic solutions is proved and the sufficient conditions for the existence of boundary periodic solutions and positive periodic solutions are derived. And then for the stochastic system, by introducing Markov regime switching, the sufficient conditions for extinction or persistence of such system are obtained. Furthermore, we proved that the system is ergodic and has a stationary distribution when the concentration of toxicant is a positive constant. Finally, two examples with numerical simulations are carried out in order to illustrate the theoretical results.

Dynamics of one-prey two-predator system with square root functional response and time lag

2015 ◽  
Vol 08 (03) ◽  
pp. 1550029 ◽  
Author(s):  
O. P. Misra ◽  
Poonam Sinha ◽  
Chhatrapal Singh
Keyword(s):  
Numerical Simulation ◽  
Functional Response ◽  
Limit Cycles ◽  
Equilibrium Points ◽  
Time Lag ◽  
Critical Value ◽  
Square Root ◽  
Basic System ◽  
Functional Responses ◽  
Predator System

Animals grouping together is one of the most interesting phenomena in population dynamics and different functional responses as a result of prey–predator forming groups have been considered by many authors in their models. In the present paper we have considered a model for one prey and two competing predator populations with time lag and square root functional response on account of herd formation by prey. It is shown that due to the inclusion of another competing predator, the underlying system without delay becomes more stable and limit cycles do not occur naturally. However, after considering the effect of time lag in the basic system, limit cycles appear in the case of all equilibrium points when delay time crosses some critical value. From the numerical simulation, it is observed that the length of delay is minimum when only prey population survives and it is maximum when all the populations coexist.

STABILITY AND BIFURCATION ANALYSIS OF A DISCRETE PREY–PREDATOR MODEL WITH SQUARE-ROOT FUNCTIONAL RESPONSE AND OPTIMAL HARVESTING

2020 ◽  
Vol 28 (01) ◽  
pp. 91-110
Author(s):  
PRABIR CHAKRABORTY ◽  
UTTAM GHOSH ◽  
SUSMITA SARKAR
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Functional Response ◽  
Population Model ◽  
Optimal Harvesting ◽  
Herd Behavior ◽  
Square Root ◽  
Optimal Harvesting Policy ◽  
Optimal Value ◽  
Predator Model

In this paper, we have considered a discrete prey–predator model with square-root functional response and optimal harvesting policy. This type of functional response is used to study the dynamics of the prey–predator model where the prey population exhibits herd behavior, i.e., the interaction between prey and predator occurs along the boundary of the population. The considered population model has three fixed points; one is trivial, the second one is axial and the last one is an interior fixed point. The first two fixed points are always feasible but the last one depends on the parameter value. The interior fixed point experiences the flip and Neimark–Sacker bifurcations depending on the predator harvesting coefficient. Finally, an optimal harvesting policy has been introduced and the optimal value of the harvesting coefficient is determined.

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