scholarly journals Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behavior

CAUCHY ◽  
2021 ◽  
Vol 6 (4) ◽  
pp. 260-269
Author(s):  
Ismail Djakaria ◽  
Muhammad Bachtiar Gaib ◽  
Resmawan Resmawan
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Equilibrium Points ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Existence And Stability ◽  
The Stability ◽  
Population Equilibrium ◽  
Predator Behavior

This paper discusses the analysis of the Rosenzweig-MacArthur predator-prey model with anti-predator behavior. The analysis is started by determining the equilibrium points, existence, and conditions of the stability. Identifying the type of Hopf bifurcation by using the divergence criterion. It has shown that the model has three equilibrium points, i.e., the extinction of population equilibrium point (E0), the non-predatory equilibrium point (E1), and the co-existence equilibrium point (E2). The existence and stability of each equilibrium point can be shown by satisfying several conditions of parameters. The divergence criterion indicates the existence of the supercritical Hopf-bifurcation around the equilibrium point E2. Finally, our model's dynamics population is confirmed by our numerical simulations by using the 4th-order Runge-Kutta methods.

scholarly journals Dynamics of an Eco-Epidemic Predator–Prey Model Involving Fractional Derivatives with Power-Law and Mittag–Leffler Kernel

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 785
Author(s):  
Hasan S. Panigoro ◽  
Agus Suryanto ◽  
Wuryansari Muharini Kusumawinahyu ◽  
Isnani Darti
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Derivative ◽  
Equilibrium Point ◽  
Power Law ◽  
Equilibrium Points ◽  
Essential Difference ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Infected Prey

In this paper, we consider a fractional-order eco-epidemic model based on the Rosenzweig–MacArthur predator–prey model. The model is derived by assuming that the prey may be infected by a disease. In order to take the memory effect into account, we apply two fractional differential operators, namely the Caputo fractional derivative (operator with power-law kernel) and the Atangana–Baleanu fractional derivative in the Caputo (ABC) sense (operator with Mittag–Leffler kernel). We take the same order of the fractional derivative in all equations for both senses to maintain the symmetry aspect. The existence and uniqueness of solutions of both eco-epidemic models (i.e., in the Caputo sense and in ABC sense) are established. Both models have the same equilibrium points, namely the trivial (origin) equilibrium point, the extinction of infected prey and predator point, the infected prey free point, the predator-free point and the co-existence point. For a model in the Caputo sense, we also show the non-negativity and boundedness of solution, perform the local and global stability analysis and establish the conditions for the existence of Hopf bifurcation. It is found that the trivial equilibrium point is a saddle point while other equilibrium points are conditionally asymptotically stable. The numerical simulations show that the solutions of the model in the Caputo sense strongly agree with analytical results. Furthermore, it is indicated numerically that the model in the ABC sense has quite similar dynamics as the model in the Caputo sense. The essential difference between the two models is the convergence rate to reach the stable equilibrium point. When a Hopf bifurcation occurs, the bifurcation points and the diameter of the limit cycles of both models are different. Moreover, we also observe a bistability phenomenon which disappears via Hopf bifurcation.

scholarly journals Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Heping Jiang ◽  
Huiping Fang ◽  
Yongfeng Wu
Keyword(s):  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability

Abstract This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

Hopf Bifurcation of a Delayed Predator–Prey Model with Nonconstant Death Rate and Constant-Rate Prey Harvesting

2018 ◽  
Vol 28 (14) ◽  
pp. 1850179 ◽  
Author(s):  
Fengrong Zhang ◽  
Xinhong Zhang ◽  
Yan Li ◽  
Changpin Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Differential Equations ◽  
Positive Constant ◽  
Death Rate ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Constant Rate ◽  
The Stability

This paper is concerned with a delayed predator–prey model with nonconstant death rate and constant-rate prey harvesting. We mainly study the impact of the time delay on the stability of positive constant solution of delayed differential equations and positive constant equilibrium of delayed diffusive differential equations, respectively. By choosing time delay [Formula: see text] as a bifurcation parameter, we show that Hopf bifurcation can occur as the time delay passes some critical values. In addition, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, some numerical simulations are carried out to depict our theoretical results.

scholarly journals A predator-prey model with cooperative hunting in the predator and group defense in the prey

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yanfei Du ◽  
Ben Niu ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Heteroclinic Orbits ◽  
Initial Population ◽  
Continuous Transition ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Cooperative Hunting ◽  
Prey Model ◽  
Existence And Stability ◽  
Group Defense

<p style='text-indent:20px;'>In this paper we propose a predator-prey model with a non-differentiable functional response in which the prey exhibits group defense and the predator exhibits cooperative hunting. There is a separatrix curve dividing the phase portrait. The species with initial population above the separatrix result in extinction of prey in finite time, and the species with initial population below it can coexist, oscillate sustainably or leave the prey surviving only. Detailed bifurcation analysis is carried out to explore the effect of cooperative hunting in the predator and aggregation in the prey on the existence and stability of the coexistence state as well as the dynamics of system. The model undergoes transcritical bifurcation, Hopf bifurcation, homoclinic (heteroclinic) bifurcation, saddle-node bifurcation, and Bogdanov-Takens bifurcation, and through numerical simulations it is found that it possesses rich dynamics including bubble loop of limit cycles, and open ended branch of periodic orbits disappearing through a homoclinic cycle or a loop of heteroclinic orbits. Also, a continuous transition of different types of Hopf branches are investigated which forms a global picture of Hopf bifurcation in the model.</p>

Stability and Hopf Bifurcation in a Predator–Prey Model with the Cost of Anti-Predator Behaviors

2019 ◽  
Vol 29 (13) ◽  
pp. 1950185 ◽  
Author(s):  
Ting Qiao ◽  
Yongli Cai ◽  
Shengmao Fu ◽  
Weiming wang
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle Oscillation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Existence And Stability ◽  
Cycle Oscillation ◽  
Predator Model ◽  
The Stability ◽  
The Cost

In this paper, we investigate the influence of anti-predator behavior in prey due to the fear of predators with a Beddington–DeAngelis prey–predator model analytically and numerically. We give the existence and stability of equilibria of the model, and provide the existence of Hopf bifurcation. In addition, we investigate the influence of the fear effect on the population dynamics of the model and find that the fear effect can not only reduce the population density of both predator and prey, but also prevent the occurrence of limit cycle oscillation and increase the stability of the system.

Hopf Bifurcation in a Delayed Diffusive Leslie–Gower Predator–Prey Model with Herd Behavior

2019 ◽  
Vol 29 (04) ◽  
pp. 1950055
Author(s):  
Fengrong Zhang ◽  
Yan Li ◽  
Changpin Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Stability ◽  
The Impact

In this paper, we consider a delayed diffusive predator–prey model with Leslie–Gower term and herd behavior subject to Neumann boundary conditions. We are mainly concerned with the impact of time delay on the stability of this model. First, for delayed differential equations and delayed-diffusive differential equations, the stability of the positive equilibrium and the existence of Hopf bifurcation are investigated respectively. It is observed that when time delay continues to increase and crosses through some critical values, a family of homogeneous and inhomogeneous periodic solutions emerge. Then, the explicit formula for determining the stability and direction of bifurcating periodic solutions are also derived by employing the normal form theory and center manifold theorem for partial functional differential equations. Finally, some numerical simulations are shown to support the analytical results.

scholarly journals Time delay induced Hopf bifurcation in a diffusive predator–prey model with prey toxicity

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ruizhi Yang ◽  
Yuxin Ma ◽  
Chiyu Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Functional Differential Equations ◽  
Positive Equilibrium ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Functional Differential ◽  
The Stability

AbstractIn this paper, we consider a diffusive predator–prey model with a time delay and prey toxicity. The effect of time delay on the stability of the positive equilibrium is studied by analyzing the eigenvalue spectrum. Delay-induced Hopf bifurcation is also investigated. By utilizing the normal form method and center manifold reduction for partial functional differential equations, the formulas for determining the property of Hopf bifurcation are given.

scholarly journals Dynamics of a diffusive predator–prey model with herd behavior

2020 ◽  
Vol 25 (1) ◽  
Author(s):  
Yan Li ◽  
Sanyun Li ◽  
Fengrong Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Positive Constant ◽  
Herd Behavior ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Constant Equilibrium ◽  
Prey Model ◽  
The Stability

This paper is devoted to considering a diffusive predator–prey model with Leslie–Gower term and herd behavior subject to the homogeneous Neumann boundary conditions. Concretely, by choosing the proper bifurcation parameter, the local stability of constant equilibria of this model without diffusion and the existence of Hopf bifurcation are investigated by analyzing the distribution of the eigenvalues. Furthermore, the explicit formula for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are also derived by applying the normal form theory. Next, we show the stability of positive constant equilibrium, the existence and stability of periodic solutions near positive constant equilibrium for the diffusive model. Finally, some numerical simulations are carried out to support the analytical results.

scholarly journals MODEL PREDATOR-PREY DENGAN DUA PREDATOR

2013 ◽  
Vol 5 (1) ◽  
pp. 43
Author(s):  
Danar Agus Nugroho ◽  
Rina Reorita
Keyword(s):  
Equilibrium Point ◽  
Brown Planthopper ◽  
Local Extinction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Long Time ◽  
The Stability ◽  
Brown Plant Hopper ◽  
Coexistence State

This paper discussed about the predator-prey model with two predators. This model is a development of the model given by Korobeinikov and Wake (1999). Dynamic behavior of the model can be determined based on the stability of the equilibrium point. The stability of the equilibrium point of predator-prey model with two predators on the general ecosystem shows that there is no coexistence state (grown in tandem) on both predators and for a long time one of the predators will lead to the local extinction even though there is no competition between the two predators. Furthermore, this model is applied to the brown plant hopper predator, mirid prey and tomcat prey. The result shows that the population of brown planthopper and both of the predators will oscillate towards a particular value with a shorter span of time. In the long term, the number of brown planthopper and mirid will be heading to the equilibrium point, while the tomcat will lead to local extinction.

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