Seasonally Perturbed Prey-Predator Ecological System with the Beddington-DeAngelis Functional Response

On the basis of the theories and methods of ecology and ordinary differential equation, a seasonally perturbed prey-predator system with the Beddington-DeAngelis functional response is studied analytically and numerically. Mathematical theoretical works have been pursuing the investigation of uniformly persistent, which depicts the threshold expression of some critical parameters. Numerical analysis indicates that the seasonality has a strong effect on the dynamical complexity and species biomass using bifurcation diagrams and Poincaré sections. The results show that the seasonality in three different parameters can give rise to rich and complex dynamical behaviors. In addition, the largest Lyapunov exponents are computed. This computation further confirms the existence of chaotic behavior and the accuracy of numerical simulation. All these results are expected to be of use in the study of the dynamic complexity of ecosystems.

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COMPLEXITY OF AN IVLEV'S PREDATOR-PREY MODEL WITH PULSE

Advances in Complex Systems ◽

10.1142/s0219525907001021 ◽

2007 ◽

Vol 10 (02) ◽

pp. 217-231 ◽

Cited By ~ 1

Author(s):

GUOPING PANG ◽

LANSUN CHEN

Keyword(s):

Numerical Simulations ◽

Functional Response ◽

Period Doubling ◽

Impulsive System ◽

Bifurcation Diagrams ◽

Bifurcation And Chaos ◽

Dynamic Complexity ◽

Predator Prey ◽

Predator Prey Model ◽

Predator System

In this paper, we investigate the extinction, permanence and dynamic complexity of the two-prey, one-predator system with Ivlev's functional response and impulsive perturbation on the predator at fixed moments. Conditions for the extinction and permanence of the system are established via the comparison theorem. Numerical simulations are carried out to explain the conclusions we obtain. Furthermore, the resulting bifurcation diagrams clearly show that the impulsive system takes on many forms of complexity including period-doubling bifurcation, period-halving bifurcation, and chaos.

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Subharmonic Bifurcations and Transition to Chaos in a Pipe Conveying Fluid under Harmonic Excitation

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.444-445.791 ◽

2013 ◽

Vol 444-445 ◽

pp. 791-795

Author(s):

Yi Xiang Geng ◽

Han Ze Liu

Keyword(s):

Chaotic Behavior ◽

Melnikov Method ◽

Harmonic Excitation ◽

Phase Portraits ◽

Pipe Conveying Fluid ◽

Bifurcation Diagrams ◽

Dynamical Behaviors ◽

Transition To Chaos ◽

Subharmonic Bifurcations ◽

Conveying Fluid

The subharmonic and chaotic behavior of a two end-fixed fluid conveying pipe whose base is subjected to a harmonic excitation are investigated. Melnikov method is applied for the system, and Melnikov criterions for subharmonic and homoclinic bifurcations are obtained analytically. The numerical simulations (including bifurcation diagrams, maximal Lyapunov exponents, phase portraits and Poincare map) confirm the analytical predictions and exhibit the complicated dynamical behaviors.

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Bifurcation and Chaotic Behavior of a Discrete-Time SIS Model

Discrete Dynamics in Nature and Society ◽

10.1155/2013/705601 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Junhong Li ◽

Ning Cui

Keyword(s):

Hopf Bifurcation ◽

Discrete Time ◽

Chaotic Behavior ◽

Euler Method ◽

Flip Bifurcation ◽

Bifurcation Diagrams ◽

Bifurcation And Chaos ◽

Sis Model ◽

Dynamical Behaviors ◽

The Rich

The discrete-time epidemic model is investigated, which is obtained using the Euler method. It is verified that there exist some dynamical behaviors in this model, such as transcritical bifurcation, flip bifurcation, Hopf bifurcation, and chaos. The numerical simulations, including bifurcation diagrams and computation of Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behaviors.

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Modeling the Role of Toxic Substances in a Phytoplankton-Toxic Phytoplankton-Zooplankton System

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.726-731.1600 ◽

2013 ◽

Vol 726-731 ◽

pp. 1600-1603

Author(s):

Jiang Lin Zhao ◽

Min Zhao

Keyword(s):

Mathematical Model ◽

Numerical Analysis ◽

Functional Response ◽

Significant Influence ◽

Toxic Substances ◽

Bifurcation Diagrams ◽

Toxic Phytoplankton ◽

Dynamical Complexity

In this paper, a mathematical model has been proposed, which consists of three variables: non-toxic phytoplankton (NTP), toxin producing phytoplankton (TPP) and zooplankton. In this model, an Monod- Haldane functional response is utilized to identify the grazing process of zooplankton due to the phytoplankton toxicity. The product of square of TPP density with square of NTP density is to depict the allelopathic influence on NTP. Numerical analysis indicates that the phytoplankton toxicity has a significant influence on the dynamical complexity and species biomass level through bifurcation diagrams. All these results are expected to be of significance in exploration of the dynamical complexity of ecosystems.

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

Mathematical Problems in Engineering ◽

10.1155/2018/8635937 ◽

2018 ◽

Vol 2018 ◽

pp. 1-15 ◽

Cited By ~ 10

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.

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Computational Study of a Primitive Life Model

Modern Physics Letters B ◽

10.1142/s0217984998000172 ◽

1998 ◽

Vol 12 (04) ◽

pp. 123-129 ◽

Cited By ~ 1

Author(s):

Mircea Andrecut

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Time Dependence ◽

Chaotic Behavior ◽

Computational Study ◽

Control Parameters ◽

Bifurcation Diagrams ◽

Partial Differential ◽

Life Model

We present a computational study of a primitive life model. The calculation involves a discrete treatment of a partial differential equation and some details of that problems are explained. We show that the investigated model is equivalent to a diffusively coupled logistic lattice. The bifurcation diagrams were calculated for different values of the control parameters. The obtained diagrams have shown that the time dependence of the population of the investigated model exhibits transitions between ordered and chaotic behavior. We have investigated also the patterns formation in this system.

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Dynamic Complexity of an Ivlev-Type Prey-Predator System with Impulsive State Feedback Control

Journal of Applied Mathematics ◽

10.1155/2012/534276 ◽

2012 ◽

Vol 2012 ◽

pp. 1-17 ◽

Cited By ~ 10

Author(s):

Chuanjun Dai ◽

Min Zhao ◽

Lansun Chen

Keyword(s):

Feedback Control ◽

Numerical Simulations ◽

State Feedback ◽

Sufficient Conditions ◽

State Feedback Control ◽

Bifurcation Diagrams ◽

Dynamic Complexity ◽

Impulsive State Feedback Control ◽

The Stability ◽

Predator System

The dynamic complexities of an Ivlev-type prey-predator system with impulsive state feedback control are studied analytically and numerically. Using the analogue of the Poincaré criterion, sufficient conditions for the existence and the stability of semitrivial periodic solutions can be obtained. Furthermore, the bifurcation diagrams and phase diagrams are investigated by means of numerical simulations, which illustrate the feasibility of the main results presented here.

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Stability in a Discrete Fractional Order Prey Predator System with Functional Response

International Journal of Advanced Research in Computer Science and Software Engineering ◽

10.23956/ijarcsse/sv7i5/0336 ◽

2017 ◽

Vol 7 (5) ◽

pp. 630-633 ◽

Cited By ~ 3

Author(s):

A. George Maria Selvam ◽

◽

R. Janagaraj ◽

Britto Jacob. S ◽

◽

...

Keyword(s):

Functional Response ◽

Fractional Order ◽

Predator System

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Chaotic Behavior in Gear System. Adaptation of Bifurcation Diagrams and Method of Statistical Mechanics.

TRANSACTIONS OF THE JAPAN SOCIETY OF MECHANICAL ENGINEERS Series C ◽

10.1299/kikaic.61.3108 ◽

1995 ◽

Vol 61 (587) ◽

pp. 3108-3115

Author(s):

Keijin Sato ◽

Sumio Yamamoto ◽

Kazutaka Yokota ◽

Toshihiro Aoki ◽

Shu Karube

Keyword(s):

Statistical Mechanics ◽

Chaotic Behavior ◽

Gear System ◽

Bifurcation Diagrams ◽

System Adaptation

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PHYSICS OF THE MIND: OPINION DYNAMICS AND DECISION MAKING PROCESSES BASED ON A BINARY NETWORK MODEL

International Journal of Modern Physics B ◽

10.1142/s0217979208050231 ◽

2008 ◽

Vol 22 (25n26) ◽

pp. 4482-4494 ◽

Cited By ~ 7

Author(s):

F. V. KUSMARTSEV ◽

KARL E. KÜRTEN

Keyword(s):

Decision Making ◽

Social Networks ◽

Chaotic Behavior ◽

Opinion Dynamics ◽

Human Mind ◽

Critical Parameters ◽

Opinion Formation ◽

Natural Catastrophes ◽

Risk Condition ◽

Binary Network

We propose a new theory of the human mind. The formation of human mind is considered as a collective process of the mutual interaction of people via exchange of opinions and formation of collective decisions. We investigate the associated dynamical processes of the decision making when people are put in different conditions including risk situations in natural catastrophes when the decision must be made very fast or at national elections. We also investigate conditions at which the fast formation of opinion is arising as a result of open discussions or public vote. Under a risk condition the system is very close to chaos and therefore the opinion formation is related to the order disorder transition. We study dramatic changes which may happen with societies which in physical terms may be considered as phase transitions from ordered to chaotic behavior. Our results are applicable to changes which are arising in various social networks as well as in opinion formation arising as a result of open discussions. One focus of this study is the determination of critical parameters, which influence a formation of stable mind, public opinion and where the society is placed “at the edge of chaos”. We show that social networks have both, the necessary stability and the potential for evolutionary improvements or self-destruction. We also show that the time needed for a discussion to take a proper decision depends crucially on the nature of the interactions between the entities as well as on the topology of the social networks.

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