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.

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 Bifurcations, Complex Behaviors, and Dynamic Transition in a Coupled Network of Discrete Predator-Prey System

2019 ◽  
Vol 2019 ◽  
pp. 1-22 ◽  
Author(s):  
Tousheng Huang ◽  
Huayong Zhang ◽  
Shengnan Ma ◽  
Ge Pan ◽  
Zhaodeng Wang ◽  
...  
Keyword(s):  
Center Manifold ◽  
Period Doubling ◽  
Flip Bifurcation ◽  
Closed Curves ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Period 2 ◽  
New Perspective ◽  
Matrix Center

The nonlinear dynamics of predator-prey systems coupled into network is an important issue in recent biological advances. In this research, we consider each node of the coupled network represents a discrete predator-prey system, and the network dynamics is investigated. By applying Jacobian matrix, center manifold theorem and bifurcation theorems, stability of fixed points, flip bifurcation and Neimark-Sacker bifurcation of the discrete predator-prey system are analyzed. Via the method of Lyapunov exponents, the nonchaos-chaos transition of the coupled network along the routes to chaos induced by bifurcations is determined. Numerical simulations are performed to demonstrate the bifurcations, various attractors and dynamic transitions of the coupled network. Via comparison, we find that the coupled network exhibits far richer and more complex behaviors than single predator-prey system, including period-doubling cascades in orbits of period-2, period-4, period-8, invariant closed curves, dynamic windows for periodic orbits and invariant curves, quasiperiodic orbits, tori, and chaotic sets. Moreover, the attractors of the coupled network show more diverse and complicated structures. These results may provide a new perspective on the predator-prey dynamics in complex networks.

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.

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

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.

Complex Dynamics of an Eco-Epidemic System with Disease in Prey Species

2021 ◽  
Vol 31 (03) ◽  
pp. 2150046
Author(s):  
Absos Ali Shaikh ◽  
Harekrishna Das ◽  
Nijamuddin Ali
Keyword(s):  
Prey Species ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Alternative Food ◽  
Predator Population ◽  
Free System ◽  
Predator Prey ◽  
Force Of Infection ◽  
Prey System ◽  
Half Saturation Constant

The objective of this study is to investigate the complex dynamics of an eco-epidemic predator–prey system where disease is transmitted in prey species and predator population is being provided with alternative food. Holling type-II functional response is taken into consideration for interaction of predator and prey species. The half saturation constant for infected prey, the growth rate of susceptible prey and force of infection play a significant role to create complex dynamics in this predator–prey system where alternative food is present. It is seen that healthy disease-free system is possible here. The system shows some important dynamics viz. stable coexistence, Hopf bifurcation, period-doubling bifurcation and chaos. The analytical results obtained from the model are justified numerically.

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.

scholarly journals Bifurcation and control for a discrete-time prey–predator model with Holling-IV functional response

2013 ◽  
Vol 23 (2) ◽  
pp. 247-261 ◽  
Author(s):  
Qiaoling Chen ◽  
Zhidong Teng ◽  
Zengyun Hu
Keyword(s):  
Functional Response ◽  
Discrete Time ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Sudden Onset ◽  
Flip Bifurcation ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Period 2 ◽  
Unstable Fixed Point

The dynamics of a discrete-time predator-prey model with Holling-IV functional response are investigated. It is shown that the model undergoes a flip bifurcation, a Hopf bifurcation and a saddle-node bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations not only exhibit our results with the theoretical analysis, but also show the complex dynamical behaviors, such as the period-3, 6, 9, 12, 20, 63, 70, 112 orbits, a cascade of period-doubling bifurcations in period-2, 4, 8, 16, quasi-periodic orbits, an attracting invariant circle, an inverse period-doubling bifurcation from the period-32 orbit leading to chaos and a boundary crisis, a sudden onset of chaos and a sudden disappearance of the chaotic dynamics, attracting chaotic sets and non-attracting sets. We also observe that when the prey is in chaotic dynamics the predator can tend to extinction or to a stable equilibrium. Specifically, we stabilize the chaotic orbits at an unstable fixed point by using OGY chaotic control.

scholarly journals Hopf bifurcation in a delayed predator-prey system with asymmetric functional response and additional food

2021 ◽  
Vol 6 (11) ◽  
pp. 12225-12244
Author(s):  
Luoyi Wu ◽  
◽  
Hang Zheng ◽  
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Functional Response ◽  
Additional Food ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Prey System ◽  
Form Theory

<abstract><p>In this paper, a delayed predator-prey system with additional food and asymmetric functional response is investigated. We discuss the local stability of equilibria and the existence of local Hopf bifurcation under the influence of the time delay. By using the normal form theory and center manifold theorem, the explicit formulas which determine the properties of bifurcating periodic solutions are obtained. Further, we prove that global periodic solutions exist after the second critical value of delay via Wu's theory. Finally, the correctness of the previous theoretical analysis is demonstrated by some numerical cases.</p></abstract>

CHAOTIC BEHAVIOR OF A PERIODICALLY FORCED PREDATOR–PREY SYSTEM WITH BEDDINGTON–DEANGELIS FUNCTIONAL RESPONSE AND IMPULSIVE PERTURBATIONS

2006 ◽  
Vol 09 (03) ◽  
pp. 209-222 ◽  
Author(s):  
SHUWEN ZHANG ◽  
DEJUN TAN ◽  
LANSUN CHEN
Keyword(s):  
Functional Response ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Periodic Variation ◽  
Periodic Forcing ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Impulsive Perturbations

The effects of periodic forcing and impulsive perturbations on the predator–prey model with Beddington–DeAngelis functional response are investigated. We assume periodic variation in the intrinsic growth rate of the prey as well as periodic constant impulsive immigration of the predator. The dynamical behavior of the system is simulated and bifurcation diagrams are obtained for different parameters. The results show that periodic forcing and impulsive perturbation can very easily give rise to complex dynamics, including quasi-periodic oscillating, a period-doubling cascade, chaos, a period-halving cascade, non-unique dynamics, and period windows.

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