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>

scholarly journals Dynamics in a Predator–Prey Model with Cooperative Hunting and Allee Effect

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3193
Author(s):  
Yanfei Du ◽  
Ben Niu ◽  
Junjie Wei
Keyword(s):  
Limit Cycle ◽  
Allee Effect ◽  
Heteroclinic Orbits ◽  
Initial Population ◽  
Form Analysis ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Cooperative Hunting ◽  
Prey Model ◽  
Two Delays

This paper deals with a diffusive predator–prey model with two delays. First, we consider the local bifurcation and global dynamical behavior of the kinetic system, which is a predator–prey model with cooperative hunting and Allee effect. For the model with weak cooperation, we prove the existence of limit cycle, and a loop of heteroclinic orbits connecting two equilibria at a threshold of conversion rate p=p#, by investigating stable and unstable manifolds of saddles. When p>p#, both species go extinct, and when p<p#, there is a separatrix. The species with initial population above the separatrix finally become extinct, and the species with initial population below it can be coexisting, oscillating sustainably, or surviving of the prey only. In the case with strong cooperation, we exhibit the complex dynamics of system, including limit cycle, loop of heteroclinic orbits among three equilibria, and homoclinic cycle with the aid of theoretical analysis or numerical simulation. There may be three stable states coexisting: extinction state, coexistence or sustained oscillation, and the survival of the prey only, and the attraction basin of each state is obtained in the phase plane. Moreover, we find diffusion may induce Turing instability and Turing–Hopf bifurcation, leaving the system with spatially inhomogeneous distribution of the species, coexistence of two different spatial-temporal oscillations. Finally, we consider Hopf and double Hopf bifurcations of the diffusive system induced by two delays: mature delay of the prey and gestation delay of the predator. Normal form analysis indicates that two spatially homogeneous periodic oscillations may coexist by increasing both delays.

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.

Bifurcation Analysis of a Delay-Induced Predator–Prey Model with Allee Effect and Prey Group Defense

2021 ◽  
Vol 31 (10) ◽  
pp. 2150158
Author(s):  
Yong Ye ◽  
Yi Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Allee Effect ◽  
Periodic Oscillation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Group ◽  
Prey Model ◽  
Group Defense ◽  
The Stability

In this paper, we establish a predator–prey model with focus on the Allee effect and prey group defense. The positivity and boundedness of the model, existence of equilibrium point, and stability change caused by Allee effect are studied. Bifurcation (transcritical bifurcation, Hopf bifurcation) analysis is discussed, and the direction of Hopf bifurcation is determined by calculating the first Lyapunov number. Then we introduce delay into the original model and consider the influence of delay on the stability of the model. By selecting delay as the bifurcation parameter, we obtain the existence conditions of Hopf bifurcation and the direction of Hopf bifurcation. Finally, we verify the theoretical analysis by numerical simulation. Considering both the Allee effect and the prey group defense, the dynamic behavior near the origin becomes more complex than only considering Allee effect or prey group defense in the model. Allee effect can bring the risk of extinction and the change of stability, and the delay effect can make the stable coexistence equilibrium unstable and lead to periodic oscillation.

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.

Dynamics of a stage-structured predator-prey model: cost and benefit of fear-induced group defense

2021 ◽  
pp. 110846
Author(s):  
Pijush Panday ◽  
Nikhil Pal ◽  
Sudip Samanta ◽  
Piotr Tryjanowski ◽  
Joydev Chattopadhyay
Keyword(s):  
Predator Prey ◽  
Cost And Benefit ◽  
Predator Prey Model ◽  
Prey Model ◽  
Group Defense
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