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.

Dynamics of a Predator–Prey Model with Holling Type II Functional Response Incorporating a Prey Refuge Depending on Both the Species

2019 ◽  
Vol 20 (1) ◽  
pp. 89-104 ◽  
Author(s):  
Hafizul Molla ◽  
Md. Sabiar Rahman ◽  
Sahabuddin Sarwardi
Keyword(s):  
Hopf Bifurcation ◽  
Uniform Boundedness ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Species ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Predator Model ◽  
The Stability ◽  
Manifold Theory

AbstractWe propose a mathematical model for prey–predator interactions allowing prey refuge. A prey–predator model is considered in the present investigation with the inclusion of Holling type-II response function incorporating a prey refuge depending on both prey and predator species. We have analyzed the system for different interesting dynamical behaviors, such as, persistent, permanent, uniform boundedness, existence, feasibility of equilibria and their stability. The ranges of the significant parameters under which the system admits a Hopf bifurcation are investigated. The system exhibits Hopf-bifurcation around the unique interior equilibrium point of the system. The explicit formula for determining the stability, direction and periodicity of bifurcating periodic solutions are also derived with the use of both the normal form and the center manifold theory. The theoretical findings of this study are substantially validated by enough numerical simulations. The ecological implications of the obtained results are discussed as well.

scholarly journals Dynamic Behaviors of a Nonautonomous Discrete Predator-Prey System Incorporating a Prey Refuge and Holling Type II Functional Response

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Yumin Wu ◽  
Fengde Chen ◽  
Wanlin Chen ◽  
Yuhua Lin
Keyword(s):  
Global Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Prey System ◽  
Type Ii Functional Response

A nonautonomous discrete predator-prey system incorporating a prey refuge and Holling type II functional response is studied in this paper. A set of sufficient conditions which guarantee the persistence and global stability of the system are obtained, respectively. Our results show that if refuge is large enough then predator species will be driven to extinction due to the lack of enough food. Two examples together with their numerical simulations show the feasibility of the main results.

Stability and Hopf Bifurcation of a Delayed Prey–Predator Model with Disease in the Predator

2019 ◽  
Vol 29 (07) ◽  
pp. 1950091 ◽  
Author(s):  
Chuangxia Huang ◽  
Hua Zhang ◽  
Jinde Cao ◽  
Haijun Hu
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Points ◽  
Mass Action ◽  
Functional Responses ◽  
Predator Species ◽  
Interior Equilibrium ◽  
Positivity Of Solutions ◽  
Predator Model ◽  
The Stability ◽  
Type Ii Functional Response

Dealing with the epidemiological prey–predator is very important for us to understand the dynamical characteristics of population models. The existing literature has shown that disease introduction into the predator group can destabilize the established prey–predator communities. In this paper, we establish a new delayed SIS epidemiological prey–predator model with the assumptions that the disease is transmitted among the predator species only and different type of predators have different functional responses, viz. the infected predator consumes the prey according to Holling type-II functional response and the susceptible predator consumes the prey following the law of mass action. The positivity of solutions, the existence of various equilibrium points, the stability and bifurcation at those equilibrium points are investigated at length. Using the incubation period as bifurcation parameter, it is observed that a Hopf bifurcation may occur around the equilibrium points when the parameter passes through some critical values. We also discuss the direction and stability of the Hopf bifurcation around the interior equilibrium point. Simulations are arranged to show the correctness and effectiveness of these theoretical results.

scholarly journals Dynamics of a Two Prey and One Predator System with Indirect Effect

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 436
Author(s):  
Renato Colucci ◽  
Érika Diz-Pita ◽  
M. Victoria Otero-Espinar
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Indirect Effect ◽  
Population Model ◽  
Type Ii ◽  
The Stability ◽  
Theoretical Results ◽  
Predator System ◽  
Type Ii Functional Response

We study a population model with two preys and one predator, considering a Holling type II functional response for the interaction between first prey and predator and taking into account indirect effect of predation. We perform the stability analysis of equilibria and study the possibility of Hopf bifurcation. We also include a detailed discussion on the problem of persistence. Several numerical simulations are provided in order to illustrate the theoretical results of the paper.

scholarly journals Global dynamics and parameter identifiability in a predator-prey interaction model

2018 ◽  
Vol 5 (1) ◽  
pp. 113-126
Author(s):  
Jai Prakash Tripathi ◽  
Suraj S. Meghwani ◽  
Swati Tyagi ◽  
Syed Abbas
Keyword(s):  
Interaction Model ◽  
Positive Equilibrium ◽  
Global Dynamics ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
Permanence Property

AbstractThis paper discusses a predator-prey model with prey refuge. We investigate the role of prey refuge on the existence and stability of the positive equilibrium. The global asymptotic stability of positive interior equilibrium solution is established using suitable Lyapunov functional, which shows that the prey refuge has no influence on the permanence property of the system. Mathematically, we analyze the effect of increase or decrease of prey reserve on the equilibrium states of prey and predator species. To access the usability of proposed predator-prey model in practical scenarios, we also suggest, the use of Levenberg-Marquardt (LM) method for associated parameter estimation problem. Numerical results demonstrate faithful reconstruction of system dynamics by estimated parameter by LM method. The analytical results found in this paper are illustrated with the help of suitable numerical examples

scholarly journals Impact of the fear and Allee effect on a Holling type II prey–predator model

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Binfeng Xie
Keyword(s):  
Hopf Bifurcation ◽  
Allee Effect ◽  
Jacobian Matrix ◽  
Positive Equilibrium ◽  
Type Ii ◽  
Bifurcation Parameter ◽  
Fear Effect ◽  
Predator Model ◽  
The Stability ◽  
Theoretical Results

AbstractIn this paper, we propose and investigate a prey–predator model with Holling type II response function incorporating Allee and fear effect in the prey. First of all, we obtain all possible equilibria of the model and discuss their stability by analyzing the eigenvalues of Jacobian matrix around the equilibria. Secondly, it can be observed that the model undergoes Hopf bifurcation at the positive equilibrium by taking the level of fear as bifurcation parameter. Moreover, through the analysis of Allee and fear effect, we find that: (i) the fear effect can enhance the stability of the positive equilibrium of the system by excluding periodic solutions; (ii) increasing the level of fear and Allee can reduce the final number of predators; (iii) the Allee effect also has important influence on the permanence of the predator. Finally, numerical simulations are provided to check the validity of the theoretical results.

scholarly journals Dynamics on Effect of Prey Refuge Proportional to Predator in Discrete-Time Prey-Predator Model

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
G. S. Mahapatra ◽  
P. K. Santra ◽  
Ebenezer Bonyah
Keyword(s):  
Period Doubling ◽  
Type Ii ◽  
Central Question ◽  
Prey Refuge ◽  
Refuge Effect ◽  
Different Types ◽  
Existence And Stability ◽  
Predator Model ◽  
Type Ii Functional Response ◽  
Prey Refuges

Prey-predator models with refuge effect have great importance in the context of ecology. Constant refuge and refuge proportional to prey are the most popular concepts of refuge in the existing literature. Now, there are new different types of refuge concepts attracting researchers. This study considers a refuge concept proportional to the predator due to the fear induced by predators. When predators increase, fears also increase and that is why prey refuges also increase. Here, we examine the influence of prey refuge proportional to predator effect in a discrete prey-predator interaction with the Holling type II functional response model. Is this refuge stabilizing or destabilizing the system? That is the central question of this study. The existence and stability of fixed points, Period-Doubling Bifurcation, Neimark–Sacker Bifurcation, the influence of prey refuge, and chaos are analyzed. This work provides the bifurcation diagrams and Lyapunov exponents to analyze the refuge parameter of the model. The proposed discrete model indicates rich dynamics as the effect of prey refuge through numerical simulations.

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 Order and Chaos in a Prey-Predator Model Incorporating Refuge, Disease, and Harvesting

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Dahlia Khaled Bahlool ◽  
Huda Abdul Satar ◽  
Hiba Abdullah Ibrahim
Keyword(s):  
Equilibrium Points ◽  
Global Dynamics ◽  
Persistence Conditions ◽  
Local Bifurcation Analysis ◽  
Uniqueness Of The Solution ◽  
Harvesting Process ◽  
Predator Model ◽  
The Stability ◽  
Predator System ◽  
Type Ii Functional Response

In this paper, a mathematical model consisting of a prey-predator system incorporating infectious disease in the prey has been proposed and analyzed. It is assumed that the predator preys upon the nonrefugees prey only according to the modified Holling type-II functional response. There is a harvesting process from the predator. The existence and uniqueness of the solution in addition to their bounded are discussed. The stability analysis of the model around all possible equilibrium points is investigated. The persistence conditions of the system are established. Local bifurcation analysis in view of the Sotomayor theorem is carried out. Numerical simulation has been applied to investigate the global dynamics and specify the effect of varying the parameters. It is observed that the system has a chaotic dynamics.

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