Global stability of a three species predator-prey food chain dynamics.

*(English)*Zbl 1408.37142Summary: In this paper controlling chaotic dynamics in a three species food chain model with modified Holling type IV functional response is proposed and studied. The system is observed to be dissipative in the positive octant. The global stability of the equilibrium points is analyzed using Routh-Hurwitz criterion and Lyapunov second method. Lyaponuv exponent and bifurcation diagrams are used to study the dynamics of the system. The effect of the death rate in the dynamics of the food chain system is discussed. Moreover, the role of intraspecific competition in the dynamics of the model is investigated theoretically and additionally numerically.

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\textit{S. J. Ali} et al., Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 26, No. 1, 39--52 (2019; Zbl 1408.37142)

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##### References:

[1] | [1] A. Elhassanien, Complex Dynamics of a Forced Discrete Leslie-Gower Type Three Species Food Chain System, Dynamics of Continuous, Discrete and Impulsive Systems, Series B, 23, (2016) 91–112. |

[2] | [2] A. Peet, P. A. Deutsch and E. Peacock-Lopez, Complex Dynamics in a Three-Level System with Intraspecies Interaction, Journal of Theoretical Biology, 232, (2005) 491-503. |

[3] | [3] A. Dhooge, W. Govaerts, Y. Kuznetsov, H. G. Meijer and B. Sauttois, New Features of Software Matconte for Bifurcation Analysis and Dynamical System, Mathematical and Computer Modelling of Dynamical System, 14 (2), (2008) 147-175. |

[4] | [4] A. Hasting and T. Powell, Chaos in Three-Species Food Chain, Ecology, 72, (1991) 896–903. |

[5] | [5] A. Priyadarshi, S. Gakkar, Dynamics of Leslie-Gower Type Generalist Predator in Tri-Trophic Food Web System, Communications in Nonlinear Science and Numerical Simulation, 18, (2013) 3202-3218. · Zbl 1328.92065 |

[6] | [6] B. W. Kooi and M. P. Boer, Chaotic Behavior of a Predator-Prey System in the Chemostat, Dynamics of Continuous, Discrete and Impulsive Systems, Series B, 10, (2003) 259–272. · Zbl 1146.34319 |

[7] | [7] C. S. Holling, The Functional Response of Predators to Prey Density and its Rule in Mimicry and Population regulation, Memoirs of the Entomological Society of Canada, 45, (1965) 3–60. |

[8] | [8] C. F.Gerald and P. O. Wheatly, Applied Numerical Analysis, New York, USA, AdisonWesley, 2004. |

[9] | [9] F. Xu, X. Shu and R. Cressman, Chaos Control and Chaos Synchronization of Fractional Order Smooth Chua’s System, Dynamics of Continuous, Discrete and Impulsive Systems, Series B, 20, (2013) 117–134. · Zbl 1268.34116 |

[10] | [10] F. J. Ayala, Experimental Invalidation of the Principle of Competitive Exclusion, Nature Publishing Group, 1969. |

[11] | [11] G. W. Harrison, Global Stability of Predator-Prey Interactions, Journal of Mathematical Biology, 8 (2), (1979) 159-171. · Zbl 0425.92009 |

[12] | [12] H. I. Freedman and S. Ruan, Uniform Persistence in Functional Differential Equations, Journal of Differential Equations, 115 (1), (1995) 173-192. · Zbl 0814.34064 |

[13] | [13] J. B. Collings, The Effect of the Functional Response on the Bifurcation Behavior of a Mite Predator-Prey Interaction Model, Journal of Mathematical Biology, 36, (1997) 149-168. · Zbl 0890.92021 |

[14] | [14] J. K. Hale, Ordinary Differential Equation, Wiley-Interscience, New York, 1969. · Zbl 0186.40901 |

[15] | [15] L. Gardini, R. Lupini. and M. G. Messia, Hopf-Bifurcation and Transition to Chaos in Lotka-Volterra Equation, Journal of Mathematical Biology, 27, (1989) 259-272. · Zbl 0715.92020 |

[16] | [16] L. JS. Allen, Introduction to mathematical biology, Pearson-Prentice Hall, NewJersey, 2007. |

[17] | [17] M. A. Aziz-Alaoui, Study of a Leslie-Gower type Tritrophic Population Model, Choas, Solitons and Fractals, 14, (2002) 1275-1293. · Zbl 1031.92027 |

[18] | [18] M. L. Rosenweig and R. H. MacArthur, Graphical Representation and Stability Conditions of Predator-Prey Interactions, American Naturalist, 97, (1963) 209-223. |

[19] | [19] R. K. Naji, R. Kumar and V. Rai, Dynamical Consequences of Predator Interference in Tritrophic Model Food Chain, Nonlinear Analysis:Real World Applications, 11, (2010) 809-818. 52S.J. Ali, N.M. Arifin, R.K. Naji, F. Ismail and N. Bachok · Zbl 1181.37120 |

[20] | [20] R. K. Upadhyay and S. N. Raw, Complex Dynamics of a Three Species Food Chain Model with Holling Type IV Functional Response, Nonlinear Analysis : Modeling and Control, 16 (3), (2011) 353-374. · Zbl 1272.49085 |

[21] | [21] R. K. Upadhyay, SRK. Iyengar, Introduction to Mathematical Modeling and Chaotic Dynamics, CRC Press, A Chapman and Hall Book, 121-122, 2014. |

[22] | [22] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, New Jersey, 1973. |

[23] | [23] S. J. Ali, N. M. Arifin, R. K. Naji, F. Ismail and N. Bachok, Analysis of Ecological Model with Holling Type IV Functional Response, International Journal of Pure and Applied Mathematics, 106 (1), (2016) 317–331. |

[24] | [24] S. J. Ali, N. M. Arifin, R. K. Naji, F. Ismail and N. Bachok, Controlling Chaotic Dynamics of a Continuous Ecological Model, International Journal of Pure and Applied Mathematics, 109 (2), (2016) 317–331. |

[25] | [25] S. J. Ali, N. M. Arifin, R. K. Naji, F. Ismail and N. Bachok, Boundedness and Stability of Leslie-Gower Model with Sokol-Howell Functional Response, Recent Advances in Mathematical Sciences, 13-26, Springer Science+Business Media, Singapore, 2016. |

[26] | [26] S. J. Ali, N. M. Arifin, R. K. Naji, F. Ismail and N. Bachok, Dynamics of Leslie-Gower Model with Simplified Holling Type IV Functional Response, Journal of Nonlinear Systems and Applications, 5 (1), (2016) 25–33. |

[27] | [27] S. Gakkhar and K. Gupta, Global Stability and Existence of Sliding Bifurcations in Filippov Type Predator-Prey Model, Dynamics of Continuous, Discrete and Impulsive Systems, Series B, 24, (2017) 1–23. · Zbl 1361.34051 |

[28] | [28] S. Ruan and D. Xiao, Global Analysis in a Predator-Prey System with Non-Monotonic Functional Response, SIAM Journal of Applied Mathematics, 61 (4), (2001) 14451472. · Zbl 0986.34045 |

[29] | [29] W. Sokol and J. A. Howell, The Kinetics of Phenol Oxidation by Washed Cells, Biotechnology and Bioengineering, 30, (1987) 921-927. |

[30] | [30] C. Xu and Z. Li, Influence of Intraspecific Density Dependence on a Three Species Food Chain with and without External Stochastic Disturbances, Ecological Modelling, 155 (1), (2002) 71-83. |

[31] | [31] S. Ellner and P. Turchin, Chaos in a Noisy World: New Methods and Evidence From Time-Series Analysis, American Naturalist, 145, (1995) 343-375. |

[32] | [32] T.C Gard and T. G. Hallam, Persistence in Food Webs-I. Lotka-Volttera Food Chains, Bulletin of Mathematical Biology, 41, (1979) 877–891. · Zbl 0422.92017 |

[33] | [33] T. Fayeldi, A. Suryanto and A. Widodo, Dynamical behaviors of a discrete SIR epidemic model with nonmonotone incidence rate, International Journal of Applied Mathematics and Statistics, 47, (2013) 416-423. |

[34] | [34] V. Rai and R. Seernivasan, Period-Doubling Bifurcations Leading to Chaos in a Model Food Chain, Ecological Modelling, 69 (1), (1993) 63–77. |

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