Complex Dynamic Behavior of a Discrete Prey–Predator–Scavenger Model with Fractional Order

2020 ◽  
Vol 17 (5) ◽  
pp. 2136-2146
Author(s):  
A. George Maria Selvam ◽  
R. Dhineshbabu ◽  
Ozlem Ak Gumus
Keyword(s):  
Fractional Order ◽  
Period Doubling ◽  
Discrete Version ◽  
Flip Bifurcation ◽  
Step Size ◽  
Dynamical Nature ◽  
The Rich ◽  
Parameter Values ◽  
The Impact ◽  
Predator System

This present work, we discuss the complex dynamical nature of the three species prey–predator system in presence of scavenger with fractional order. A discretization process is applied to obtain its discrete version and the impact of the scavenger on the ecosystem is discussed. Firstly, existence and local stability of the equilibria state of the model are addressed. The presences of Flip bifurcation and Neimark-Sacker bifurcation around boundary and coexistence equilibrium states are studied by using bifurcation theory. The trajectories and phase plane diagrams are plotted for biologically meaningful sets of parameter values. Numerical simulations for the effect of step size and fractional order parameters of the system are performed, on the first part to demonstrate the analytical results and on the second part revealing new motivating the dynamics behaviors of the scavenger model including period-doubling cascades (flip), periods—2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24 and 32 orbits, invariant cycles, a chaotic attractors, and the rich dynamic behaviors of the discretized model is exhibited.

scholarly journals Bifurcation and Chaos Analysis for a Discrete Ecological Developmental Systems

2021 ◽  
Author(s):  
Xiaowei Jiang ◽  
Chaoyang Chen ◽  
Xian-He Zhang ◽  
Ming Chi ◽  
Huaicheng Yan
Keyword(s):  
Discrete System ◽  
Period Doubling ◽  
Discrete Systems ◽  
Flip Bifurcation ◽  
Maximum Lyapunov Exponent ◽  
Step Size ◽  
Developmental Systems ◽  
Bifurcation And Chaos ◽  
The Rich ◽  
Theoretical Results

Abstract This work concentrates on the dynamic analysis including bifurcation and chaos of a discrete ecological developmental systems. Specifically, it is a prey-predator-scavenger (PPS) system, which is derived by Euler discretization method. By choosing the step size h as a bifurcation parameter, we determine the set consist of all system’s parameters, in which the system can undergo flip bifurcation (FB) and Neimark-Sacker bifurcation (NSB). The theoretical results are verified by some numerical simulations. It is shown that the discrete systems exhibit more interesting behaviors, including the chaotic sets, quasiperiodic orbits, and the cascade of period-doubling bifurcation in orbits of periods-2, 4, 8, 16. Finally, corresponding to the two bifurcation behaviors discussed, the maximum Lyapunov exponent is numerically calculated, which further verifies the rich dynamic characteristics of the discrete system.

scholarly journals Stability, Bifurcation, Chaos : Discrete Prey Predator Model with Step Size

2019 ◽  
Vol 9 (1) ◽  
pp. 3382-3387 ◽  
Keyword(s):  
Chaotic Behavior ◽  
Initial Conditions ◽  
Equilibrium States ◽  
Phase Portraits ◽  
Step Size ◽  
Dynamical Nature ◽  
Predator Model ◽  
The Stability ◽  
Parameter Values ◽  
Bifurcation Chaos

In this work titled Stability, Bifurcation, Chaos: Discrete prey predator model with step size, by Forward Euler Scheme method the discrete form is obtained. Equilibrium states are calculated and the stability of the equilibrium states and dynamical nature of the model are examined in the closed first quadrant 2 R with the help of variation matrix. It is observed that the system is sensitive to the initial conditions and also to parameter values. The dynamical nature of the model is investigated with the assistance of Lyapunov Exponent, bifurcation diagrams, phase portraits and chaotic behavior of the system is identified. Numerical simulations validate the theoretical observations.

A fractional-order ship power system: chaos and its dynamical properties

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Karthikeyan Rajagopal ◽  
Prakash Duraisamy ◽  
Goitom Tadesse ◽  
Christos Volos ◽  
Fahimeh Nazarimehr ◽  
...  
Keyword(s):  
Power System ◽  
Fractional Order ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Order Form ◽  
Dynamical Behaviors ◽  
Fractional Order Controller ◽  
Parameter Values ◽  
Ship Power System ◽  
D Model

Abstract In this research, the ship power system is studied with a fractional-order approach. A 2-D model of a two-generator parallel-connected is considered. A chaotic attractor is observed for particular parameter values. The fractional-order form is calculated with the Adam–Bashforth–Moulton method. The chaotic response is identified even for the order 0.99. Phase portrait is generated using the Caputo derivative approach. Wolf’s algorithm is used to calculate Lyapunov exponents. For the considered values of parameters, one positive Lyapunov exponent confirms the existence of chaos. Bifurcation diagrams are presented to analyze the various dynamical behaviors and bifurcation points. Interestingly, the considered system is multistable. Also, antimonotonicity, period-doubling, and period halving are observed in the bifurcation diagram. As the last step, a fractional-order controller is designed to remove chaotic dynamics. Time plots are simulated to show the effectiveness of the controller.

scholarly journals Stability Results for Two-Dimensional Systems of Fractional-Order Difference Equations

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1751
Author(s):  
Oana Brandibur ◽  
Eva Kaslik ◽  
Dorota Mozyrska ◽  
Małgorzata Wyrwas
Keyword(s):  
Asymptotic Stability ◽  
Fractional Order ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Discrete Version ◽  
Model Parameters ◽  
Step Size ◽  
Order Difference ◽  
Stability And Instability ◽  
Fractional Orders

Linear autonomous incommensurate systems that consist of two fractional-order difference equations of Caputo-type are studied in terms of their asymptotic stability and instability properties. More precisely, the asymptotic stability of the considered linear system is fully characterized, in terms of the fractional orders of the considered Caputo-type differences, as well as the elements of the linear system’s matrix and the discretization step size. Moreover, fractional-order-independent sufficient conditions are also derived for the instability of the system under investigation. With the aim of exemplifying the theoretical results, a fractional-order discrete version of the FitzHugh–Nagumo neuronal model is constructed and analyzed. Furthermore, numerical simulations are undertaken in order to substantiate the theoretical findings, showing that the membrane potential may exhibit complex bursting behavior for suitable choices of the model parameters and fractional orders of the Caputo-type differences.

Complex Dynamics of a Discrete-Time Prey–Predator System with Leslie Type: Stability, Bifurcation Analyses and Chaos

2020 ◽  
Vol 30 (10) ◽  
pp. 2050149
Author(s):  
Pinar Baydemir ◽  
Huseyin Merdan ◽  
Esra Karaoglu ◽  
Gokce Sucu
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Point ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Positive Equilibrium ◽  
Flip Bifurcation ◽  
Step Size ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Predator System

Dynamic behavior of a discrete-time prey–predator system with Leslie type is analyzed. The discrete mathematical model was obtained by applying the forward Euler scheme to its continuous-time counterpart. First, the local stability conditions of equilibrium point of this system are determined. Then, the conditions of existence for flip bifurcation and Neimark–Sacker bifurcation arising from this positive equilibrium point are investigated. More specifically, by choosing integral step size as a bifurcation parameter, these bifurcations are driven via center manifold theorem and normal form theory. Finally, numerical simulations are performed to support and extend the theoretical results. Analytical results show that an integral step size has a significant role on the dynamics of a discrete system. Numerical simulations support that enlarging the integral step size causes chaotic behavior.

Bifurcation, Chaos and Turing Instability Analysis for a Space-Time Discrete Toxic Phytoplankton-Zooplankton Model with Self-Diffusion

2019 ◽  
Vol 29 (13) ◽  
pp. 1950184
Author(s):  
Shihong Zhong ◽  
Jinliang Wang ◽  
You Li ◽  
Nan Jiang
Keyword(s):  
Numerical Simulations ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Turing Instability ◽  
Space Time ◽  
Discrete Version ◽  
Flip Bifurcation ◽  
Center Manifold Theorem ◽  
Toxic Phytoplankton ◽  
Stable Conditions

The spatiotemporal dynamics of a space-time discrete toxic phytoplankton-zooplankton model is studied in this paper. The stable conditions for steady states are obtained through the linear stability analysis. According to the center manifold theorem and bifurcation theory, the critical parameter values for flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are determined, respectively. Besides, the numerical simulations are provided to illustrate theoretical results. In order to distinguish chaos from regular behaviors, the maximum Lyapunov exponents are shown. The simulations show new and complex dynamics behaviors, such as period-doubling cascade, invariant circles, periodic windows, chaotic region and pattern formations. Numerical simulations of Turing patterns induced by flip-Turing instability, Neimark–Sacker Turing instability and chaos reveal a variety of spatiotemporal patterns, including plaque, curl, spiral, circle, and many other regular and irregular patterns. In comparison with former results in literature, the space-time discrete version considered in this paper captures more complicated and richer nonlinear dynamics behaviors and contributes a new comprehension on the complex pattern formation of spatially extended discrete phytoplankton-zooplankton system.

Bifurcation and Chaos in a Discrete Fractional Order Prey-Predator System Involving Infection in Prey

2020 ◽  
pp. 95-119
Author(s):  
A. George Maria Selvam ◽  
R. Dhineshbabu
Keyword(s):  
Fractional Order ◽  
Control Method ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Order System ◽  
Qualitative Behavior ◽  
Uniqueness Of Solutions ◽  
Predator System

This chapter considers the dynamical behavior of a new form of fractional order three-dimensional continuous time prey-predator system and its discretized counterpart. Existence and uniqueness of solutions is obtained. The dynamic nature of the model is discussed through local stability analysis of the steady states. Qualitative behavior of the model reveals rich and complex dynamics as exhibited by the discrete-time fractional order model. Moreover, the bifurcation theory is applied to investigate the presence of Neimark-Sacker and period-doubling bifurcations at the coexistence steady state taking h as a bifurcation parameter for the discrete fractional order system. Also, the trajectories, phase diagrams, limit cycles, bifurcation diagrams, and chaotic attractors are obtained for biologically meaningful sets of parameter values for the discretized system. Finally, the analytical results are strengthened with appropriate numerical examples and they demonstrate the chaotic behavior over a range of parameters. Chaos control is achieved by the hybrid control method.

scholarly journals Bifurcations and Synchronization of the Fractional-Order Bloch System

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Honggang Dang ◽  
Lixin Yang
Keyword(s):  
Fractional Order ◽  
Limit Cycles ◽  
System Parameter ◽  
Period Doubling ◽  
Adaptive Synchronization ◽  
Fractional Order Systems ◽  
Unknown Parameters ◽  
The Rich ◽  
Period Limit ◽  
The Stability

In this paper, bifurcations and synchronization of a fractional-order Bloch system are studied. Firstly, the bifurcations with the variation of every order and the system parameter for the system are discussed. The rich dynamics in the fractional-order Bloch system including chaos, period, limit cycles, period-doubling, and tangent bifurcations are found. Furthermore, based on the stability theory of fractional-order systems, the adaptive synchronization for the system with unknown parameters is realized by designing appropriate controllers. Numerical simulations are carried out to demonstrate the effectiveness and flexibility of the controllers.

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

2017 ◽  
Vol 7 (5) ◽  
pp. 630-633 ◽  
Author(s):  
A. George Maria Selvam ◽  
◽  
R. Janagaraj ◽  
Britto Jacob. S ◽  
◽  
...  
Keyword(s):  
Functional Response ◽  
Fractional Order ◽  
Predator System

Heterotrophic Kinetic Parameter Estimation for Enhanced Biological Phosphorus Removal Processes Operated in Conventional and Membrane-Assisted Modes

2006 ◽  
Vol 41 (1) ◽  
pp. 72-83 ◽  
Author(s):  
Zhe Zhang ◽  
Eric R. Hall
Keyword(s):  
Parameter Estimation ◽  
Phosphorus Removal ◽  
Growth Yield ◽  
Pilot Scale ◽  
Enhanced Biological Phosphorus Removal ◽  
Biological Phosphorus Removal ◽  
Nitrogen And Phosphorus ◽  
Parameter Values ◽  
The Impact ◽  
Biological Phosphorus

Abstract Parameter estimation and wastewater characterization are crucial for modelling of the membrane enhanced biological phosphorus removal (MEBPR) process. Prior to determining the values of a subset of kinetic and stoichiometric parameters used in ASM No. 2 (ASM2), the carbon, nitrogen and phosphorus fractions of influent wastewater at the University of British Columbia (UBC) pilot plant were characterized. It was found that the UBC wastewater contained fractions of volatile acids (SA), readily fermentable biodegradable COD (SF) and slowly biodegradable COD (XS) that fell within the ASM2 default value ranges. The contents of soluble inert COD (SI) and particulate inert COD (XI) were somewhat higher than ASM2 default values. Mixed liquor samples from pilot-scale MEBPR and conventional enhanced biological phosphorus removal (CEBPR) processes operated under parallel conditions, were then analyzed experimentally to assess the impact of operation in a membrane-assisted mode on the growth yield (YH), decay coefficient (bH) and maximum specific growth rate of heterotrophic biomass (µH). The resulting values for YH, bH and µH were slightly lower for the MEBPR train than for the CEBPR train, but the differences were not statistically significant. It is suggested that MEBPR simulation using ASM2 could be accomplished satisfactorily using parameter values determined for a conventional biological phosphorus removal process, if MEBPR parameter values are not available.

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