scholarly journals Complex Dynamics of a Discrete-Time Predator-Prey System with Ivlev Functional Response

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Hunki Baek
Keyword(s):  
Functional Response ◽  
Discrete Time ◽  
Complex Dynamics ◽  
Discrete Systems ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Dynamical Complexity ◽  
Dynamical Behaviors ◽  
Prey System

The dynamics of a discrete-time predator-prey system with Ivlev functional response is investigated in this paper. The conditions of existence for flip bifurcation and Hopf bifurcation in the interior of R+2 are derived by using the center manifold theorem and bifurcation theory. Numerical simulations are presented not only to substantiate our theoretical results but also to illustrate the complex dynamical behaviors of the system such as attracting invariant circles, periodic-doubling bifurcation leading to chaos, and periodic-halving phenomena. In addition, the maximum Lyapunov exponents are numerically calculated to confirm the dynamical complexity of the system. Finally, we compare the system to discrete systems with Holling-type functional response with respect to dynamical behaviors.

scholarly journals Complex Dynamics on the Routes to Chaos in a Discrete Predator-Prey System with Crowley-Martin Type Functional Response

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
Huayong Zhang ◽  
Shengnan Ma ◽  
Tousheng Huang ◽  
Xuebing Cong ◽  
Zichun Gao ◽  
...  
Keyword(s):  
Functional Response ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Period Doubling ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Routes To Chaos ◽  
Prey System ◽  
The One

We present in this paper an investigation on a discrete predator-prey system with Crowley-Martin type functional response to know its complex dynamics on the routes to chaos which are induced by bifurcations. Via application of the center manifold theorem and bifurcation theorems, occurrence conditions for flip bifurcation and Neimark-Sacker bifurcation are determined, respectively. Numerical simulations are performed, on the one hand, verifying the theoretical results and, on the other hand, revealing new interesting dynamical behaviors of the discrete predator-prey system, including period-doubling cascades, period-2, period-3, period-4, period-5, period-6, period-7, period-8, period-9, period-11, period-13, period-15, period-16, period-20, period-22, period-24, period-30, and period-34 orbits, invariant cycles, chaotic attractors, sub-flip bifurcation, sub-(inverse) Neimark-Sacker bifurcation, chaotic interior crisis, chaotic band, sudden disappearance of chaotic dynamics and abrupt emergence of chaos, and intermittent periodic behaviors. Moreover, three-dimensional bifurcation diagrams are utilized to study the transition between flip bifurcation and Neimark-Sacker bifurcation, and a critical case between the two bifurcations is found. This critical bifurcation case is a combination of flip bifurcation and Neimark-Sacker bifurcation, showing the nonlinear characteristics of both bifurcations.

Complex dynamics of a discrete-time predator-prey system with Holling IV functional response

2016 ◽  
Vol 87 ◽  
pp. 158-171 ◽  
Author(s):  
Qianqian Cui ◽  
Qiang Zhang ◽  
Zhipeng Qiu ◽  
Zengyun Hu
Keyword(s):  
Functional Response ◽  
Discrete Time ◽  
Complex Dynamics ◽  
Predator Prey ◽  
Prey System

scholarly journals Hopf Bifurcation and Global Periodic Solutions in a Predator-Prey System with Michaelis-Menten Type Functional Response and Two Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Yunxian Dai ◽  
Yiping Lin ◽  
Huitao Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Center Manifold ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Prey System ◽  
Model Study ◽  
Two Delays ◽  
Normal Form Method

We consider a predator-prey system with Michaelis-Menten type functional response and two delays. We focus on the case with two unequal and non-zero delays present in the model, study the local stability of the equilibria and the existence of Hopf bifurcation, and then obtain explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation when the delaysτ1≠τ2.

scholarly journals Dynamical behaviors in a discrete fractional-order predator-prey system

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5857-5874 ◽  
Author(s):  
Yao Shi ◽  
Qiang Ma ◽  
Xiaohua Ding
Keyword(s):  
Numerical Simulations ◽  
Fixed Points ◽  
Fractional Order ◽  
Control Strategies ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Local Asymptotic Stability ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Theoretical Results

This paper is related to the dynamical behaviors of a discrete-time fractional-order predatorprey model. We have investigated existence of positive fixed points and parametric conditions for local asymptotic stability of positive fixed points of this model. Moreover, it is also proved that the system undergoes Flip bifurcation and Neimark-Sacker bifurcation for positive fixed point. Various chaos control strategies are implemented for controlling the chaos due to Flip and Neimark-Sacker bifurcations. Finally, numerical simulations are provided to verify theoretical results. These results of numerical simulations demonstrate chaotic behaviors over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behaviors in the model.

Complex dynamic behaviors of a discrete predator–prey model with stage structure and harvesting

2016 ◽  
Vol 10 (01) ◽  
pp. 1750013 ◽  
Author(s):  
Boshan Chen ◽  
Jiejie Chen
Keyword(s):  
Numerical Simulations ◽  
Period Doubling ◽  
Stage Structure ◽  
Model Parameters ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model

First, a discrete stage-structured and harvested predator–prey model is established, which is based on a predator–prey model with Type III functional response. Then theoretical methods are used to investigate existence of equilibria and their local properties. Third, it is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of [Formula: see text], by using the normal form of discrete systems, the center manifold theorem and the bifurcation theory, as varying the model parameters in some range. In particular, the direction and the stability of the flip bifurcation and the Neimark–Sacker bifurcation are showed. Finally, numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascades of period-doubling bifurcation and chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors. In addition, we show also the stabilizing effect of the harvesting by using numerical simulations.

scholarly journals Bifurcation Analysis and Chaos Control in a Discrete-Time Predator-Prey System of Leslie Type with Simplified Holling Type IV Functional Response

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
S. M. Sohel Rana ◽  
Umme Kulsum
Keyword(s):  
Functional Response ◽  
Discrete Time ◽  
Chaos Control ◽  
Phase Portraits ◽  
Closed Cycle ◽  
Predator Prey ◽  
Type Iv ◽  
Prey System ◽  
Period 2 ◽  
Maximum Lyapunov Exponents

The dynamic behavior of a discrete-time predator-prey system of Leslie type with simplified Holling type IV functional response is examined. We algebraically show that the system undergoes a bifurcation (flip or Neimark-Sacker) in the interior of R+2. Numerical simulations are presented not only to validate analytical results but also to show chaotic behaviors which include bifurcations, phase portraits, period 2, 4, 6, 8, 10, and 20 orbits, invariant closed cycle, and attracting chaotic sets. Furthermore, we compute numerically maximum Lyapunov exponents and fractal dimension to justify the chaotic behaviors of the system. Finally, a strategy of feedback control is applied to stabilize chaos existing in the system.

Complex dynamics of diffusive predator–prey system with Beddington–DeAngelis functional response: The role of prey-taxis

2017 ◽  
Vol 10 (03) ◽  
pp. 1750047 ◽  
Author(s):  
Nilesh Kumar Thakur ◽  
Rashi Gupta ◽  
Ranjit Kumar Upadhyay
Keyword(s):  
Functional Response ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Spatial Models ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Active Movement ◽  
Predator Prey ◽  
Prey System

An attempt has been made to understand the complex dynamics of a spatial predator–prey system with Beddington–DeAngelis type functional response in the presence of prey-taxis and subjected to homogenous Neumann boundary condition. To describe the active movement of predators to the regions of high prey density or if the predator is following some sort of odor to find the prey, the prey-taxis phenomenon is included in a general reaction–diffusion equation. We have studied the linear stability analysis of both spatial and non-spatial models. We have performed extensive simulations to identify the conditions to generate spatiotemporal patterns in the presence of prey-taxis. It has been observed that the increasing predator active movement from the bifurcation value, the system shows chaotic behavior whereas increasing value of random movement brings the system back to order from the disordered state.

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