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 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.

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.

Bifurcation Analysis in a Lotka-Volterra Model with Delay

2012 ◽  
Vol 594-597 ◽  
pp. 2693-2696
Author(s):  
Chang Jin Xu
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Volterra Model ◽  
The Stability ◽  
Theoretical Results

In this paper, a Lotka-Volterra model with time delay is considered. The stability of the equilibrium of the model is investigated and the existence of Hopf bifurcation is proved. Numerical simulations are performed to justify the theoretical results. Finally, main conclusions are included.

STABILITY AND HOPF BIFURCATION OF A DELAYED NETWORK OF FOUR NEURONS WITH A SHORT-CUT CONNECTION

2008 ◽  
Vol 18 (10) ◽  
pp. 3053-3072 ◽  
Author(s):  
XIAOCHEN MAO ◽  
HAIYAN HU
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Center Manifold ◽  
Network Equilibrium ◽  
Stability Switches ◽  
Center Manifold Reduction ◽  
The Stability ◽  
Manifold Reduction ◽  
Theoretical Results ◽  
Good Agreement

This paper reveals the dynamics of a delayed neural network of four neurons, with a short-cut connection through a theoretical analysis and some case studies of both numerical simulations and experiments. It presents a detailed analysis of the stability and the stability switches of the network equilibrium, as well as the Hopf bifurcation and the bifurcating periodic responses on the basis of the normal form and the center manifold reduction. Afterwards, the study focuses on the validation of theoretical results through numerical simulations and circuit experiments. The numerical simulations and the circuit experiments not only show good agreement with theoretical results, but also show abundant effects of the short-cut connection on the network dynamics.

Influence of the Fear Effect on a Holling Type II Prey–Predator System with a Michaelis–Menten Type Harvesting

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Binfeng Xie ◽  
Zhengce Zhang ◽  
Na Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Oscillation ◽  
Positive Equilibrium ◽  
Type Ii ◽  
Bifurcation Parameter ◽  
Fear Effect ◽  
Existence And Stability ◽  
Positive Equilibrium Point ◽  
The Stability ◽  
Predator System

In this work, a prey–predator system with Holling type II response function including a Michaelis–Menten type capture and fear effect is put forward to be studied. Firstly, the existence and stability of equilibria of the system are discussed. Then, by considering the harvesting coefficient as bifurcation parameter, the occurrence of Hopf bifurcation at the positive equilibrium point and the existence of limit cycle emerging through Hopf bifurcation are proved. Furthermore, through the analysis of fear effect and capture item, we find that: (i) the fear effect can either stabilize the system by excluding periodic solutions or destroy the stability of the system and produce periodic oscillation behavior; (ii) increasing the level of fear can reduce the final number of predators, but not lead to extinction; (iii) the harvesting coefficient also has significant influence on the persistence of the predator. Finally, numerical simulations are presented to illustrate the results.

HOPF BIFURCATION OF A DELAYED DIFFERENTIAL EQUATION

2007 ◽  
Vol 17 (04) ◽  
pp. 1367-1374 ◽  
Author(s):  
QIAN GUO ◽  
CHANGPIN LI
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Nonlinear Differential Equation ◽  
Trivial Solution ◽  
The Stability ◽  
Analytical Expressions ◽  
Theoretical Results

In this paper, we study Hopf bifurcation of a second-order nonlinear differential equation with time delay by using the Lyapunov–Schmidt reduction. The approximate analytical expressions of the periodic solutions bifurcated from the trivial solution are given. We also discuss the stability of these periodic solutions. The numerical simulations line up with the theoretical results.

Stability and Bifurcation in a State-Dependent Delayed Predator–Prey System

2016 ◽  
Vol 26 (04) ◽  
pp. 1650060 ◽  
Author(s):  
Aiyu Hou ◽  
Shangjiang Guo
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Local Stability ◽  
Perturbation Methods ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Prey System ◽  
State Dependent ◽  
The Stability ◽  
Theoretical Results

In this paper, we consider a class of predator–prey equations with state-dependent delayed feedback. Firstly, we investigate the local stability of the positive equilibrium and the existence of the Hopf bifurcation. Then we use perturbation methods to determine the sub/supercriticality of Hopf bifurcation and hence the stability of Hopf bifurcating periodic solutions. Finally, numerical simulations supporting our theoretical results are also provided.

scholarly journals Hopf Bifurcation and Hybrid Control of a Delayed Ecoepidemiological Model with Nonlinear Incidence Rate and Holling Type II Functional Response

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Miao Peng ◽  
Zhengdi Zhang ◽  
C. W. Lim ◽  
Xuedi Wang
Keyword(s):  
Hopf Bifurcation ◽  
Incidence Rate ◽  
Functional Response ◽  
Hybrid Control ◽  
Type Ii ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Center Manifold Theorem ◽  
The Stability ◽  
Type Ii Functional Response

Hopf bifurcation analysis of a delayed ecoepidemiological model with nonlinear incidence rate and Holling type II functional response is investigated. By analyzing the corresponding characteristic equations, the conditions for the stability and existence of Hopf bifurcation for the system are obtained. In addition, a hybrid control strategy is proposed to postpone the onset of an inherent bifurcation of the system. By utilizing normal form method and center manifold theorem, the explicit formulas that determine the direction of Hopf bifurcation and the stability of bifurcating period solutions of the controlled system are derived. Finally, some numerical simulation examples confirm that the hybrid controller is efficient in controlling Hopf bifurcation.

Predator-dependent transmissible disease spreading in prey under Holling type-II functional response

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Dipankar Ghosh ◽  
Prasun K. Santra ◽  
Abdelalim A. Elsadany ◽  
Ghanshaym S. Mahapatra
Keyword(s):  
Functional Response ◽  
Prey Species ◽  
Linearization Method ◽  
Logistic Growth ◽  
Type Ii ◽  
Disease Spreading ◽  
Predator Interactions ◽  
The Stability ◽  
Infected Prey ◽  
Type Ii Functional Response

Abstract This paper focusses on developing two species, where only prey species suffers by a contagious disease. We consider the logistic growth rate of the prey population. The interaction between susceptible prey and infected prey with predator is presumed to be ruled by Holling type II and I functional response, respectively. A healthy prey is infected when it comes in direct contact with infected prey, and we also assume that predator-dependent disease spreads within the system. This research reveals that the transmission of this predator-dependent disease can have critical repercussions for the shaping of prey–predator interactions. The solution of the model is examined in relation to survival, uniqueness and boundedness. The positivity, feasibility and the stability conditions of the fixed points of the system are analysed by applying the linearization method and the Jacobian matrix method.

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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