scholarly journals Fear effect in discrete prey-predator model incorporating square root functional response

2021 ◽  
Vol 2 (2) ◽  
pp. 51-57
Author(s):  
P.K. Santra
Keyword(s):  
Functional Response ◽  
Stability Condition ◽  
Discrete Model ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Square Root ◽  
Bifurcation Diagrams ◽  
Fear Effect ◽  
Hypothetical Data ◽  
Predator Model

In this work, an interaction between prey and its predator involving the effect of fear in presence of the predator and the square root functional response is investigated. Fixed points and their stability condition are calculated. The conditions for the occurrence of some phenomena namely Neimark-Sacker, Flip, and Fold bifurcations are given. Base on some hypothetical data, the numerical simulations consist of phase portraits and bifurcation diagrams are demonstrated to picturise the dynamical behavior. It is also shown numerically that rich dynamics are obtained by the discrete model as the effect of fear.

scholarly journals Discrete-time prey-predator model with θ-logistic growth for prey incorporating square root functional response

2020 ◽  
Vol 1 (2) ◽  
pp. 41-48
Author(s):  
P.K. Santra
Keyword(s):  
Functional Response ◽  
Discrete Time ◽  
Discrete Model ◽  
Phase Portraits ◽  
Logistic Growth ◽  
Herd Behavior ◽  
Square Root ◽  
Existence And Stability ◽  
Predator Model ◽  
Predator System

This article presents the dynamics of a discrete-time prey-predator system with square root functional response incorporating θ-logistic growth. This type of functional response is used to study the dynamics of the prey--predator system where the prey population exhibits herd behavior, i.e., the interaction between prey and predator occurs along the boundary of the population. The existence and stability of fixed points and Neimark-Sacker Bifurcation (NSB) are analyzed. The phase portraits, bifurcation diagrams and Lyapunov exponents are presented and analyzed for different parameters of the model. Numerical simulations show that the discrete model exhibits rich dynamics as the effect of θ-logistic growth.

STABILITY AND BIFURCATION ANALYSIS OF A DISCRETE PREY–PREDATOR MODEL WITH SQUARE-ROOT FUNCTIONAL RESPONSE AND OPTIMAL HARVESTING

2020 ◽  
Vol 28 (01) ◽  
pp. 91-110
Author(s):  
PRABIR CHAKRABORTY ◽  
UTTAM GHOSH ◽  
SUSMITA SARKAR
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Functional Response ◽  
Population Model ◽  
Optimal Harvesting ◽  
Herd Behavior ◽  
Square Root ◽  
Optimal Harvesting Policy ◽  
Optimal Value ◽  
Predator Model

In this paper, we have considered a discrete prey–predator model with square-root functional response and optimal harvesting policy. This type of functional response is used to study the dynamics of the prey–predator model where the prey population exhibits herd behavior, i.e., the interaction between prey and predator occurs along the boundary of the population. The considered population model has three fixed points; one is trivial, the second one is axial and the last one is an interior fixed point. The first two fixed points are always feasible but the last one depends on the parameter value. The interior fixed point experiences the flip and Neimark–Sacker bifurcations depending on the predator harvesting coefficient. Finally, an optimal harvesting policy has been introduced and the optimal value of the harvesting coefficient is determined.

scholarly journals Bifurcation Diagrams and Global Phase Portraits for Some Hamiltonian Systems with Rational Potentials

2018 ◽  
Vol 28 (13) ◽  
pp. 1850168
Author(s):  
Ting Chen ◽  
Jaume Llibre
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Normal Forms ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Linear Change ◽  
Rational Potential ◽  
Change Of Variables ◽  
Poincaré Disk

In this paper, we study the global dynamical behavior of the Hamiltonian system [Formula: see text], [Formula: see text] with the rational potential Hamiltonian [Formula: see text], where [Formula: see text] and [Formula: see text] are polynomials of degree 1 or 2. First we get the normal forms for these rational Hamiltonian systems by some linear change of variables. Then we classify all the global phase portraits of these systems in the Poincaré disk and provide their bifurcation diagrams.

Bifurcations in a Seasonally Forced Predator–Prey Model with Generalized Holling Type IV Functional Response

2016 ◽  
Vol 26 (12) ◽  
pp. 1650203 ◽  
Author(s):  
Jingli Ren ◽  
Xueping Li
Keyword(s):  
Periodic Solutions ◽  
Functional Response ◽  
Complex Dynamics ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Chaotic Motions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Numerical Bifurcation

A seasonally forced predator–prey system with generalized Holling type IV functional response is considered in this paper. The influence of seasonal forcing on the system is investigated via numerical bifurcation analysis. Bifurcation diagrams for periodic solutions of periods one and two, containing bifurcation curves of codimension one and bifurcation points of codimension two, are obtained by means of a continuation technique, corresponding to different bifurcation cases of the unforced system illustrated in five bifurcation diagrams. The seasonally forced model exhibits more complex dynamics than the unforced one, such as stable and unstable periodic solutions of various periods, stable and unstable quasiperiodic solutions, and chaotic motions through torus destruction or cascade of period doublings. Finally, some phase portraits and corresponding Poincaré map portraits are given to illustrate these different types of solutions.

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 Codimension-2 Bifurcation Analysis and Control of a Discrete Mosquito Model with a Proportional Release Rate of Sterile Mosquitoes

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Qiaoling Chen ◽  
Zhidong Teng ◽  
Junli Liu ◽  
Feng Wang
Keyword(s):  
Release Rate ◽  
Discrete Model ◽  
Bifurcation Theory ◽  
Control Strategies ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Dynamical Behaviors ◽  
And Control ◽  
Theoretical Results ◽  
Maximum Lyapunov Exponents

This paper concerns a discrete wild and sterile mosquito model with a proportional release rate of sterile mosquitoes. It is shown that the discrete model undergoes codimension-2 bifurcations with 1 : 2, 1 : 3, and 1 : 4 strong resonances by applying the bifurcation theory. Some numerical simulations, including codimension-2 bifurcation diagrams, maximum Lyapunov exponents diagrams, and phase portraits, are also presented to illustrate the validity of theoretical results and display the complex dynamical behaviors. Moreover, two control strategies are applied to the model.

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