scholarly journals Mathematical Analysis of a Fractional-Order Predator-Prey Network with Feedback Control Strategy

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Wei Zhang ◽  
Yu Fei ◽  
Zhouhong Li ◽  
Chengdai Huang
Keyword(s):  
Fractional Order ◽  
Bifurcation Control ◽  
Theoretical Derivation ◽  
Predator Prey ◽  
Stability Performance ◽  
Proposed Model ◽  
Predator Prey Models ◽  
The Stability ◽  
Feedback Control Strategy ◽  
Theoretical Results

This paper examines the bifurcation control problem of a class of delayed fractional-order predator-prey models in accordance with an enhancing feedback controller. Firstly, the bifurcation points of the devised model are precisely figured out via theoretical derivation taking time delay as a bifurcation parameter. Secondly, a set comparative analysis on the influence of bifurcation control is numerically studied containing enhancing feedback, dislocated feedback, and eliminating feedback approaches. It can be seen that the stability performance of the proposed model can be immensely heightened by the enhancing feedback approach. At the end, a numerical example is given to illustrate the feasibility of the theoretical results.

Dynamical Behaviors of a Fractional-Order Predator–Prey Model with Holling Type IV Functional Response and Its Discretization

2019 ◽  
Vol 20 (2) ◽  
pp. 125-136
Author(s):  
A. M. Yousef ◽  
S. Z. Rida ◽  
Y. Gh. Gouda ◽  
A. S. Zaki
Keyword(s):  
Functional Response ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Dynamical Behaviors ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Theoretical Results

AbstractIn this paper, we investigate the dynamical behaviors of a fractional-order predator–prey with Holling type IV functional response and its discretized counterpart. First, we seek the local stability of equilibria for the fractional-order model. Also, the necessary and sufficient conditions of the stability of the discretized model are achieved. Bifurcation types (include transcritical, flip and Neimark–Sacker) and chaos are discussed in the discretized system. Finally, numerical simulations are executed to assure the validity of the obtained theoretical results.

An extended optimal velocity difference model in a cooperative driving system

2015 ◽  
Vol 26 (05) ◽  
pp. 1550054
Author(s):  
Jinliang Cao ◽  
Zhongke Shi ◽  
Jie Zhou
Keyword(s):  
Traffic Flow ◽  
Linear Stability Theory ◽  
Difference Model ◽  
Optimal Velocity ◽  
Driving System ◽  
Cooperative Driving ◽  
Proposed Model ◽  
De Vries ◽  
The Stability ◽  
Theoretical Results

An extended optimal velocity (OV) difference model is proposed in a cooperative driving system by considering multiple OV differences. The stability condition of the proposed model is obtained by applying the linear stability theory. The results show that the increase in number of cars that precede and their OV differences lead to the more stable traffic flow. The Burgers, Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations are derived to describe the density waves in the stable, metastable and unstable regions, respectively. To verify these theoretical results, the numerical simulation is carried out. The theoretical and numerical results show that the stabilization of traffic flow is enhanced by considering multiple OV differences. The traffic jams can be suppressed by taking more information of cars ahead.

Pattern Formation Controlled by External Forcing in a Spatial Harvesting Predator-Prey Model

2011 ◽  
pp. 214-221
Author(s):  
Feng Rao
Keyword(s):  
Reaction Diffusion ◽  
Euler Method ◽  
External Forcing ◽  
Periodic Forcing ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Flux Boundary Conditions ◽  
Predator Prey Models ◽  
The Stability

Predator–prey models in ecology serve a variety of purposes, which range from illustrating a scientific concept to representing a complex natural phenomenon. Due to the complexity and variability of the environment, the dynamic behavior obtained from existing predator–prey models often deviates from reality. Many factors remain to be considered, such as external forcing, harvesting and so on. In this chapter, we study a spatial version of the Ivlev-type predator-prey model that includes reaction-diffusion, external periodic forcing, and constant harvesting rate on prey. Using this model, we study how external periodic forcing affects the stability of predator-prey coexistence equilibrium. The results of spatial pattern analysis of the Ivlev-type predator-prey model with zero-flux boundary conditions, based on the Euler method and via numerical simulations in MATLAB, show that the model generates rich dynamics. Our results reveal that modeling by reaction-diffusion equations with external periodic forcing and nonzero constant prey harvesting could be used to make general predictions regarding predator-prey equilibrium,which may be used to guide management practice, and to provide a basis for the development of statistical tools and testable hypotheses.

scholarly journals Stability Anomalies of Some Jacobian-Free Iterative Methods of High Order of Convergence

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 51
Author(s):  
Alicia Cordero ◽  
Javier G. Maimó ◽  
Juan R. Torregrosa ◽  
María P. Vassileva
Keyword(s):  
Difference Operator ◽  
Divided Difference ◽  
Polynomial System ◽  
Nonlinear Problems ◽  
Coefficient Matrix ◽  
Test Problems ◽  
Stability Performance ◽  
Need To Evaluate ◽  
The Stability ◽  
Theoretical Results

In this manuscript, we design two classes of parametric iterative schemes to solve nonlinear problems that do not need to evaluate Jacobian matrices and need to solve three linear systems per iteration with the same divided difference operator as the coefficient matrix. The stability performance of the classes is analyzed on a quadratic polynomial system, and it is shown that for many values of the parameter, only convergence to the roots of the problem exists. Finally, we check the performance of these methods on some test problems to confirm the theoretical results.

scholarly journals Correction: FRACTIONAL ORDER BARBALAT’S LEMMA AND ITS APPLICATIONS IN THE STABILITY OF FRACTIONAL ORDER NONLINEAR SYSTEMS

2017 ◽  
Vol 22 (4) ◽  
pp. 503-513 ◽  
Author(s):  
Fei Wang ◽  
Yongqing Yang
Keyword(s):  
Nonlinear Systems ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Caputo Fractional Derivative ◽  
Barbalat’S Lemma ◽  
Liouville Derivative ◽  
Liouville Fractional Derivative ◽  
The Stability ◽  
The Relationship ◽  
Theoretical Results

This paper investigates fractional order Barbalat’s lemma and its applications for the stability of fractional order nonlinear systems with Caputo fractional derivative at first. Then, based on the relationship between Caputo fractional derivative and Riemann-Liouville fractional derivative, fractional order Barbalat’s lemma with Riemann-Liouville derivative is derived. Furthermore, according to these results, a set of new formulations of Lyapunov-like lemmas for fractional order nonlinear systems are established. Finally, an example is presented to verify the theoretical results in this paper.

scholarly journals Theoretical Studies on the Effects of Dispersal Corridors on the Permanence of Discrete Predator-Prey Models in Patchy Environment

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Chunqing Wu ◽  
Shengming Fan ◽  
Patricia J. Y. Wong
Keyword(s):  
Sufficient Conditions ◽  
Theoretical Studies ◽  
Patchy Environment ◽  
Predator Prey ◽  
Numerical Examples ◽  
Predator Prey Model ◽  
Prey Model ◽  
Dispersal Corridors ◽  
Predator Prey Models ◽  
Theoretical Results

We study two discrete predator-prey models in patchy environment, one without dispersal corridors and one with dispersal corridors. Dispersal corridors are passes that allow the migration of species from one patch to another and their existence may influence the permanence of the model. We will offer sufficient conditions to guarantee the permanence of the two predator-prey models. By comparing the two permanence criteria, we discuss the effects of dispersal corridors on the permanence of the predator-prey model. It is found that the dispersion of the prey from one patch to another is helpful to the permanence of the prey if the population growth of the prey is density dependent; however, this dispersion of the prey could be disadvantageous or advantageous to the permanence of the predator. Five numerical examples are presented to confirm the theoretical results obtained and to illustrate the effects of dispersal corridors on the permanence of the predator-prey model.

scholarly journals Adaptive - Synchronization of Fractional-Order Chaotic Systems with Nonidentical Structures

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Li-xin Yang ◽  
Wan-sheng He
Keyword(s):  
Adaptive Control ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Chaotic Systems ◽  
Control Technique ◽  
Adaptive Synchronization ◽  
Fractional Order Systems ◽  
The Stability ◽  
Different Chaotic Systems ◽  
Theoretical Results

This paper investigates the adaptive - synchronization of the fractional-order chaotic systems with nonidentical structures. Based on the stability of fractional-order systems and adaptive control technique, a general formula for designing the controller and parameters update law is proposed to achieve adaptive - synchronization between two different chaotic systems with different structures. The effective scheme parameters identification and - synchronization of chaotic systems can be realized simultaneously. Furthermore, two typical illustrative numerical simulations are given to demonstrate the effectiveness of the proposed scheme, for each case, we design the controller and parameter update laws in detail. The numerical simulations are performed to verify the effectiveness of the theoretical 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 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.

scholarly journals Deep-Reinforcement Learning-Based Co-Evolution in a Predator–Prey System

Entropy ◽  
2019 ◽  
Vol 21 (8) ◽  
pp. 773 ◽  
Author(s):  
Xueting Wang ◽  
Jun Cheng ◽  
Lei Wang
Keyword(s):  
Reinforcement Learning ◽  
Large Scale ◽  
Behavior Pattern ◽  
Learning Ability ◽  
Predator Prey ◽  
Complex Processes ◽  
Proposed Model ◽  
The Stability ◽  
Reasonable Behavior ◽  
Evolutionary Networks

Understanding or estimating the co-evolution processes is critical in ecology, but very challenging. Traditional methods are difficult to deal with the complex processes of evolution and to predict their consequences on nature. In this paper, we use the deep-reinforcement learning algorithms to endow the organism with learning ability, and simulate their evolution process by using the Monte Carlo simulation algorithm in a large-scale ecosystem. The combination of the two algorithms allows organisms to use experiences to determine their behavior through interaction with that environment, and to pass on experience to their offspring. Our research showed that the predators’ reinforcement learning ability contributed to the stability of the ecosystem and helped predators obtain a more reasonable behavior pattern of coexistence with its prey. The reinforcement learning effect of prey on its own population was not as good as that of predators and increased the risk of extinction of predators. The inconsistent learning periods and speed of prey and predators aggravated that risk. The co-evolution of the two species had resulted in fewer numbers of their populations due to their potentially antagonistic evolutionary networks. If the learnable predators and prey invade an ecosystem at the same time, prey had an advantage. Thus, the proposed model illustrates the influence of learning mechanism on a predator–prey ecosystem and demonstrates the feasibility of predicting the behavior evolution in a predator–prey ecosystem using AI approaches.

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