scholarly journals Dynamics of a discrete-time system with Z-type control

2020 ◽  
Vol 75 (7) ◽  
pp. 609-620
Author(s):  
Shilpa Garai ◽  
Moumita Garain ◽  
Sudip Samanta ◽  
Nikhil Pal
Keyword(s):  
Discrete Time ◽  
Control Mechanism ◽  
Population Control ◽  
Supporting Evidence ◽  
Discrete Time System ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Set Up ◽  
The Stability

AbstractIn community ecology, the stability of a predator–prey system is a considerably desired issue; as a result, population control of a predator–prey system is very important. The dynamics of continuous-time models with Z-type control is studied earlier. But, the effectiveness of the Z-type control mechanism in a discrete-time set-up is lacking. First, we consider a Lotka–Volterra type discrete-time predator–prey model. We observe that without control, the system exhibits rich dynamical behaviors including chaotic oscillations. We apply the Z-control mechanism in both direct and indirect ways to the system and observe that in both cases, controllers have the property to drive the populations of the system to the desired state. We conduct numerical simulation as supporting evidence of our analytical results.

scholarly journals Further Results on Bifurcation for a Fractional-Order Predator-Prey System concerning Mixed Time Delays

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Zhenjiang Yao ◽  
Bingnan Tang
Keyword(s):  
Fractional Order ◽  
Time Delays ◽  
Bifurcation Theory ◽  
Predator Prey ◽  
Mixed Time Delays ◽  
Predator Prey Model ◽  
Prey System ◽  
Change Of Variable ◽  
Set Up ◽  
The Stability

In the present work, we mainly focus on a new established fractional-order predator-prey system concerning both types of time delays. Exploiting an advisable change of variable, we set up an isovalent fractional-order predator-prey model concerning a single delay. Taking advantage of the stability criterion and bifurcation theory of fractional-order dynamical system and regarding time delay as bifurcation parameter, we establish a new delay-independent stability and bifurcation criterion for the involved fractional-order predator-prey system. The numerical simulation figures and bifurcation plots successfully support the correctness of the established key conclusions.

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.

scholarly journals Discrete-Time Predator-Prey Interaction with Selective Harvesting and Predator Self-Limitation

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Zhihua Chen ◽  
Qamar Din ◽  
Muhammad Rafaqat ◽  
Umer Saeed ◽  
Muhammad Bilal Ajaz
Keyword(s):  
Discrete Time ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
State Feedback Control ◽  
Time Model ◽  
Predator Prey ◽  
Selective Harvesting ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Interaction

Selective harvesting plays an important role on the dynamics of predator-prey interaction. On the other hand, the effect of predator self-limitation contributes remarkably to the stabilization of exploitative interactions. Keeping in view the selective harvesting and predator self-limitation, a discrete-time predator-prey model is discussed. Existence of fixed points and their local dynamics is explored for the proposed discrete-time model. Explicit principles of Neimark–Sacker bifurcation and period-doubling bifurcation are used for discussion related to bifurcation analysis in the discrete-time predator-prey system. The control of chaotic behavior is discussed with the help of methods related to state feedback control and parameter perturbation. At the end, some numerical examples are presented for verification and illustration of theoretical findings.

Global Stability of a Non-Smooth Predator–Prey System with Holling I Functional Response and Refuge Effect

2014 ◽  
Vol 595 ◽  
pp. 283-288 ◽  
Author(s):  
Yuan Tian ◽  
Hai Ting Sun ◽  
Yu Xia He
Keyword(s):  
Global Stability ◽  
Functional Response ◽  
Tangent Point ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Refuge Effect ◽  
Prey System ◽  
Trivial Equilibrium ◽  
The Stability ◽  
Regular Equilibrium

This paper analyses the dynamics of a non-smooth predator-prey model with refuge effect, where the functional response is taken as Holling I type. To begin with, some preliminaries and the existence of regular, virtual, pseudo-equilibrium and tangent point are established. Then, the stability of trivial equilibrium and predator free equilibrium is discussed. Furthermore, it is shown that the regular equilibrium and the pseudo-equilibrium cannot coexist. Finally, the conclusion is given.

scholarly journals Stability analysis of an eco-epidemiological model incorporating a prey refuge

2010 ◽  
Vol 15 (4) ◽  
pp. 473-491 ◽  
Author(s):  
A. K. Pal ◽  
G. P. Samanta
Keyword(s):  
Stability Analysis ◽  
Time Delay ◽  
Computer Simulations ◽  
Discrete Time ◽  
Predator Population ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Discrete Time Delay ◽  
The Stability

The present paper deals with the problem of a predator-prey model incorporating a prey refuge with disease in the prey-population. We assume the predator population will prefer only infected population for their diet as those are more vulnerable. Dynamical behaviours such as boundedness, permanence, local and global stabilities are addressed. We have also studied the effect of discrete time delay on the model. The length of delay preserving the stability is also estimated. Computer simulations are carried out to illustrate our analytical findings.

scholarly journals Time-delay effect on a diffusive predator–prey model with habitat complexity

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yanfeng Li ◽  
Haicheng Liu ◽  
Ruizhi Yang
Keyword(s):  
Time Delay ◽  
Habitat Complexity ◽  
Functional Differential Equations ◽  
Neumann Boundary ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Functional Differential ◽  
The Stability

AbstractBased on the predator–prey system with a Holling type functional response function, a diffusive predator–prey system with digest delay and habitat complexity is proposed. Firstly, the stability of the equilibrium of diffusion system without delay is studied. Secondly, under the Neumann boundary conditions, taking time delay as the bifurcation parameter, by analyzing the eigenvalues of linearized operator of the system and using the normal form theory and center manifold method of partial functional differential equations, the effect of time delay on the stability of the system is studied and the conditions under which Hopf bifurcation occurs are given. In addition, the calculation formulas of the bifurcation direction and the stability of bifurcating periodic solutions are derived. Finally, the accuracy of theoretical analysis results is verified by numerical simulations and the biological explanation is given for the analysis results.

Chaos control in a discrete-time predator–prey model with weak Allee effect

2018 ◽  
Vol 11 (07) ◽  
pp. 1850089 ◽  
Author(s):  
Saheb Pal ◽  
Sourav Kumar Sasmal ◽  
Nikhil Pal
Keyword(s):  
Discrete Time ◽  
Allee Effect ◽  
Sufficient Conditions ◽  
Equilibrium Analysis ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Stability ◽  
The Impact

The stability of the predator–prey model subject to the Allee effect is an interesting topic in recent times. In this paper, we investigate the impact of weak Allee effect on the stability of a discrete-time predator–prey model with Holling type-IV functional response. The mathematical features of the proposed model are analyzed with the help of equilibrium analysis, stability analysis, and bifurcation theory. We provide sufficient conditions for the flip bifurcation by considering Allee parameter as the bifurcation parameter. We observe that the model becomes stable from chaotic dynamics as the Allee parameter increases. Further, we observe bi-stability behavior of the model between only prey existence equilibrium and the coexistence equilibrium. Our analytical findings are illustrated through numerical simulations.

scholarly journals Global Stability and Hopf Bifurcation for Gause-Type Predator-Prey System

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Shuang Guo ◽  
Weihua Jiang
Keyword(s):  
Global Stability ◽  
Periodic Solutions ◽  
Three Dimensional ◽  
Growth Time ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
System With Time Delay

A class of three-dimensional Gause-type predator-prey model is considered. Firstly, local stability of equilibrium indicating the extinction of top-predator is obtained. Meanwhile, we construct a Lyapunov function, which is an extension of the Lyapunov functions constructed by Hsu for predator-prey system (2005), to give the global stability of the equilibrium. Secondly, we analyze the stability of coexisting equilibrium of predator-prey system with time delay when the predator catches the prey of pregnancy or with growth time. The delay can lead to periodic solutions, which is consistent with the law of growth for birds and some mammals. Further, an explicit formula is given which determines the stability of the bifurcating periodic solutions theoretically and the existence of periodic solutions is displayed by numerical simulations.

scholarly journals Stability and Hopf Bifurcation of a Predator-Prey Model with Distributed Delays and Competition Term

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Lv-Zhou Zheng
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Distributed Delays ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Positive Equilibrium Point ◽  
The Stability

A class of predator-prey system with distributed delays and competition term is considered. By considering the time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the predator-prey system. According to the theorem of Hopf bifurcation, some sufficient conditions are obtained for the local stability of the positive equilibrium point.

STABILITY ANALYSIS OF A PREDATOR-PREY MODEL

2012 ◽  
Vol 05 (01) ◽  
pp. 1250007 ◽  
Author(s):  
ZHICHAO JIANG ◽  
ZHAOZHUANG GUO ◽  
YUEFANG SUN
Keyword(s):  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Center Manifold Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Characteristic Values ◽  
The Stability ◽  
Manifold Theory

In this paper, a time-delayed predator-prey system is considered. The existence of Hopf bifurcations at the positive equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. Numerical simulations to support the analytical conclusions are carried out.

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