Bifurcation Based-Delay Feedback Control Strategy for a Fractional-Order Two-Prey One-Predator System

The issue of bifurcation control for a novel fractional-order two-prey and one-predator system with time delay is dealt with in this paper. Firstly, the characteristic equation is investigated by picking time delay as the bifurcation parameter, and some conditions for the appearance of Hopf bifurcation are obtained. It is shown that time delay can give rise to periodic oscillations and each order has an important impact on the occurrence of Hopf bifurcation for the controlled system. Then, it is illustrated that the control result is obviously influenced by the feedback gain. It is also noted that the inception of the bifurcation can be postponed if the feedback gain decreases. Finally, two simulation examples are carried out to verify the chief theoretical results.

[1]  R. Rakkiyappan,et al.  Fractional-order delayed predator–prey systems with Holling type-II functional response , 2015 .

[2]  Jing Chen,et al.  Bifurcations of Invariant Tori in Predator-Prey Models with Seasonal Prey Harvesting , 2013, SIAM J. Appl. Math..

[3]  E. Ahmed,et al.  On fractional order differential equations model for nonlocal epidemics , 2007, Physica A: Statistical Mechanics and its Applications.

[4]  Coșkun Yakar,et al.  Stability, Boundedness, and Lagrange Stability of Fractional Differential Equations with Initial Time Difference , 2014, TheScientificWorldJournal.

[5]  Jinde Cao,et al.  New bifurcation results for fractional BAM neural network with leakage delay , 2017 .

[6]  Sze-Bi Hsu,et al.  Analysis of three species Lotka-Volterra food web models with omnivory , 2015 .

[7]  Andrea Baronchelli,et al.  Machine Learning the Cryptocurrency Market , 2018, Complex..

[8]  Kehui Sun,et al.  Complex Dynamics of the Fractional-Order Rössler System and Its Tracking Synchronization Control , 2018, Complex..

[9]  Xuebing Zhang,et al.  Dynamic analysis of a fractional order delayed predator–prey system with harvesting , 2016, Theory in Biosciences.

[10]  Wendi Wang,et al.  A predator-prey system with stage-structure for predator , 1997 .

[11]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[12]  Junwei Lu,et al.  Stability and bifurcation of a delayed generalized fractional-order prey-predator model with interspecific competition , 2019, Appl. Math. Comput..

[13]  Wang Xin.  Hopf Bifurcation of a Two-prey One-predator System with Time Delays , 2008 .

[14]  Rajivganthi Chinnathambi,et al.  Stability of fractional-order prey–predator system with time-delay and Monod–Haldane functional response , 2018 .

[15]  Jinde Cao,et al.  Time-delayed feedback control of dynamical small-world networks at Hopf bifurcation , 2009 .

[16]  Jinhu Lü,et al.  Stability analysis of linear fractional differential system with multiple time delays , 2007 .

[17]  G. P. Samanta,et al.  Dynamical Behaviour of a Two Prey and One Predator System , 2014 .

[18]  A. B. Roy,et al.  Persistence of two prey-one predator system with ratio-dependent predator influence , 2000 .

[19]  Jinde Cao,et al.  Controlling bifurcation in a delayed fractional predator-prey system with incommensurate orders , 2017, Appl. Math. Comput..

[20]  Nikhil Pal,et al.  Stability and Bifurcation Analysis of a Three-Species Food Chain Model with Fear , 2015, Int. J. Bifurc. Chaos.

[21]  G. Mora,et al.  Characterization and generation of α-dense curves , 1997 .

[22]  Saudi Arabia,et al.  Two-prey one-predator model , 2007 .

[23]  S. Bhalekar,et al.  Synchronization of incommensurate non-identical fractional order chaotic systems using active control , 2014 .

[24]  Ahmed M. A. El-Sayed,et al.  On the fractional-order logistic equation , 2007, Appl. Math. Lett..

[25]  Alan A. Berryman,et al.  The Orgins and Evolution of Predator‐Prey Theory , 1992 .

[26]  Shigui Ruan,et al.  Bifurcations in a predator–prey system of Leslie type with generalized Holling type III functional response☆ , 2014 .

[27]  Rong Yuan,et al.  Stability and bifurcation in a harvested one-predator–two-prey model with delays , 2006 .

[28]  S B Saila,et al.  Interspecific competition, predation and species diversity. , 1970, Journal of theoretical biology.

[29]  Norihiko Adachi,et al.  Existence and bifurcation of stable equilibrium in two-prey, one-predator communities , 1983 .

[30]  Sergei Petrovskii,et al.  An ecological study of a marine plankton community based on the field data collected from Bay of Bengal , 2006 .

[31]  Cao-Chuan Ma,et al.  Hopf bifurcation and stability in predator-prey model with a stage-structure for prey , 2013, Appl. Math. Comput..

[32]  Zhidong Teng,et al.  Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge , 2016, Journal of Applied Mathematics and Computing.

[33]  Tamer Nabil,et al.  Fractional-Order Model of Two-Prey One-Predator System , 2017 .

[34]  M. F. Elettreby,et al.  Two-prey one-predator model , 2009 .

[35]  Jinde Cao,et al.  Modeling, Analysis and Bifurcation Control of a Delayed Fractional-Order Predator-Prey Model , 2018, Int. J. Bifurc. Chaos.

[36]  Wei Zhao,et al.  The Fractional Kalman Filter Based Asynchronous Multi-Rate Sensors Information Fusion , 2018, 2018 10th International Conference on Modelling, Identification and Control (ICMIC).

[37]  Robert M. May,et al.  Time‐Delay Versus Stability in Population Models with Two and Three Trophic Levels , 1973 .

[38]  Zaihua Wang,et al.  Stability Switches of a Class of Fractional-Delay Systems With Delay-Dependent Coefficients , 2018, Journal of Computational and Nonlinear Dynamics.