Harvesting and Hopf Bifurcation in a Prey-Predator Model with Holling Type IV Functional Response

2012 ◽  
Vol 2 (1) ◽  
pp. 99 ◽  
Author(s):  
Manju Agarwal ◽  
Rachana Pathak
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Type Iv ◽  
Predator Model

scholarly journals Bifurcation Analysis of a Delayed Predator-Prey Model with Holling Type III Functional Response and Predator Harvesting

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Uttam Das ◽  
T. K. Kar
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Normal Form Theory ◽  
Predator Species ◽  
Center Manifold Theorem ◽  
Type Iii ◽  
Predator Prey Model ◽  
Form Theory ◽  
Predator Model ◽  
The Stability

This paper tries to highlight a delayed prey-predator model with Holling type III functional response and harvesting to predator species. In this context, we have discussed local stability of the equilibria, and the occurrence of Hopf bifurcation of the system is examined by considering the harvesting effort as bifurcation parameter along with the influences of harvesting effort of the system when time delay is zero. Direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also studied by applying the normal form theory and the center manifold theorem. Lastly some numerical simulations are carried out to draw for the validity of the theoretical results.

scholarly journals A mathematical analysis of Hopf-bifurcation in a prey-predator model with nonlinear functional response

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Assane Savadogo ◽  
Boureima Sangaré ◽  
Hamidou Ouedraogo
Keyword(s):  
Numerical Simulation ◽  
Hopf Bifurcation ◽  
Functional Response ◽  
Mathematical Analysis ◽  
Existence And Uniqueness ◽  
Nonlinear Functional ◽  
Uniqueness Of The Solution ◽  
Nontrivial Periodic Solutions ◽  
Predator Model ◽  
Stability Results

AbstractIn this paper, our aim is mathematical analysis and numerical simulation of a prey-predator model to describe the effect of predation between prey and predator with nonlinear functional response. First, we develop results concerning the boundedness, the existence and uniqueness of the solution. Furthermore, the Lyapunov principle and the Routh–Hurwitz criterion are applied to study respectively the local and global stability results. We also establish the Hopf-bifurcation to show the existence of a branch of nontrivial periodic solutions. Finally, numerical simulations have been accomplished to validate our analytical findings.

scholarly journals Bifurkasi Hopf pada model prey-predator-super predator dengan fungsi respon yang berbeda

2020 ◽  
Vol 1 (2) ◽  
pp. 65-70
Author(s):  
Dian Savitri ◽  
Hasan S. Panigoro
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Diagram ◽  
Functional Response ◽  
Conversion Rate ◽  
Phase Portraits ◽  
Type I ◽  
Existence And Stability ◽  
Predator Model ◽  
The One ◽  
Extinction Point

This article discusses the one-prey, one-predator, and the super predator model with different types of functional response. The rate of prey consumption by the predator follows Holling type I functional response and the rate of predator consumption by the super predator follows Holling type II functional response. We identify the existence and stability of critical points and obtain that the extinction of all population points is always unstable, and the other two are conditionally stable i.e., the super predator extinction point and the co-existence point. Furthermore, we give the numerical simulations to describe the bifurcation diagram and phase portraits of the model. The bifurcation diagram is obtained by varying the parameter of the conversion rate of predator biomass into a new super-predator which gives forward and Hopf bifurcation. The forward bifurcation occurs around the super predator extinction point while Hopf bifurcation occurs around the interior of the model. Based on the terms of existence and numerical simulation, we confirm that the conversion rate of predator biomass into a new super-predator controls the dynamics of the system and maintains the existence of predator.

scholarly journals Stability Analysis of Prey-Predator Model With Holling Type IV Functional Response and Infectious Predator

2020 ◽  
Vol 17 (2) ◽  
pp. 155-165
Author(s):  
A. Muh. Amil Siddik ◽  
Syamsuddin Toaha ◽  
Andi Muhammad Anwar
Keyword(s):  
Stability Analysis ◽  
Equilibrium Point ◽  
Functional Response ◽  
Local Stability ◽  
Jacobian Matrix ◽  
Equilibrium Points ◽  
Type Iv ◽  
Simulation Results ◽  
Predator Model ◽  
Parameter Values

Stability of equilibrium points of the prey-predator model with diseases that spreads in predators where the predation function follows the simplified Holling type IV functional response are investigated. To find out the local stability of the equilibrium point of the model, the system is then linearized around the equilibrium point using the Jacobian matrix method, and stability of the equilibrium point is determined via the eigenvalues method. There exists three non-negative equilibrium points, except , that may exist and stable. Simulation results show that with the variation of several parameter values infection rate of disease , the diseases in the system may become endemic, or may become free from endemic.  

Dynamical Behaviors of a Delayed Prey–Predator Model with Beddington–DeAngelis Functional Response: Stability and Periodicity

2020 ◽  
Vol 30 (16) ◽  
pp. 2050244
Author(s):  
Xin Zhang ◽  
Renxiang Shi ◽  
Ruizhi Yang ◽  
Zhangzhi Wei
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Functional Response ◽  
Discrete Time ◽  
Numerical Continuation ◽  
Positive Equilibrium ◽  
Local Hopf Bifurcation ◽  
Discrete Time Delay ◽  
Predator Model ◽  
The Stability

This work investigates a prey–predator model with Beddington–DeAngelis functional response and discrete time delay in both theoretical and numerical ways. Firstly, we incorporate into the system a discrete time delay between the capture of the prey by the predator and its conversion to predator biomass. Moreover, by taking the delay as a bifurcation parameter, we analyze the stability of the positive equilibrium in the delayed system. We analytically prove that the local Hopf bifurcation critical values are neatly paired, and each pair is joined by a bounded global Hopf branch. Also, we show that the predator becomes extinct with an increase of the time delay. Finally, before the extinction of the predator, we find the abundance of dynamical complexity, such as supercritical Hopf bifurcation, using the numerical continuation package DDE-BIFTOOL.

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