Permanence for a delayed discrete three-level food-chain model with Beddington-DeAngelis functional response.

*(English)*Zbl 1120.92049From the paper: The dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Two species predator-prey models have been extensively investigated in the literature, but recently more and more attention has been focused on systems with three or more trophic levels, since the use of simple food chain models can assist qualitatively illustrating the complexity and interdependencies in real ecological systems.

Until very recently, both ecologists and mathematicians chose to base their studies on the Beddington-DeAngelis functional response, which has some of the same qualitative features as the ratio-dependent form but avoids some of the singular behaviors of ratio-dependent models at low densities which have been the source of controversy and can provide better description of predator feeding over a range of predator-prey abundances; this is stronlgy supported by numerous field and laboratory experiments and observations. Here, a discrete three-level food-chain model with Beddington-DeAngelis functional response is investigated. It is shown that the system is permanent under some appropriate conditions.

Until very recently, both ecologists and mathematicians chose to base their studies on the Beddington-DeAngelis functional response, which has some of the same qualitative features as the ratio-dependent form but avoids some of the singular behaviors of ratio-dependent models at low densities which have been the source of controversy and can provide better description of predator feeding over a range of predator-prey abundances; this is stronlgy supported by numerous field and laboratory experiments and observations. Here, a discrete three-level food-chain model with Beddington-DeAngelis functional response is investigated. It is shown that the system is permanent under some appropriate conditions.

##### MSC:

92D40 | Ecology |

39A11 | Stability of difference equations (MSC2000) |

39A12 | Discrete version of topics in analysis |

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\textit{C. Xu} and \textit{M. Wang}, Appl. Math. Comput. 187, No. 2, 1109--1119 (2007; Zbl 1120.92049)

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##### References:

[1] | Hsu, S.B.; Hwang, T.W.; Kuang, Y., A ratio-dependent food chain model and its applications to biological control, Math. biosci., 181, 55-83, (2003) · Zbl 1036.92033 |

[2] | Beddington, J.R., Mutual interference between parasites or predators and its effects on searching efficiency, J. animal ecol., 44, 331-340, (1975) |

[3] | DeAngelis, D.L.; Goldstein, R.A.; ONeill, R.V., A model for trophic interaction, Ecology, 56, 881-892, (1975) |

[4] | Raid Kamel Naji, Alla Tariq Balasim, Dynamic behavior of a three food chain model with Beddington-DeAngelis functional response, Chaos Solutions & Fractals, in press. · Zbl 1195.92061 |

[5] | Huo, Hai-Feng; Li, Wan-tong, Existence and global stability of a periodic food chain model with delays, Appl. math. comput., 162, 1333-1349, (2005) · Zbl 1073.93050 |

[6] | Sun, Y.G.; Saker, S.H., Positive periodic solutions of discrete three-level food-chain model of Holling type, Appl. math. comput., 180, 353-365, (2006) · Zbl 1099.92079 |

[7] | Xu, Rui; Chen, Lansun, Periodic solutions of a discrete time lotka – volterra type food-chain model with delays, Appl. math. comput., 171, 91-103, (2005) · Zbl 1081.92043 |

[8] | Zhang, Huiying; Chen, Fengde, Positive periodic solutions for three-species food chain system with beddington – deangelis functional response, J. fuzhou univ., 32, 413-416, (2004), (in Chinese) · Zbl 1092.34560 |

[9] | Fan, Yong-Hong; Li, Wan-tong, Permanence for a delayed discrete ratio-dependent predator – prey system with Holling type functional response, J. math. anal. appl., 299, 357-374, (2004) · Zbl 1063.39013 |

[10] | Wang, L.; Wang, M.Q., Ordinary difference equations, (1989), Xinjiang University Press Xinjiang, (in Chinese) |

[11] | Kocic, V.L.; Ladas, G., Global behavior of nonlinear difference equations of higher order with applications, (1993), Kluwer Academic London · Zbl 0787.39001 |

[12] | Hwang, T.W., Global analysis of the predator – prey system with beddington – deangelis functional response, J. math. anal. appl., 290, 113-122, (2004) · Zbl 1086.34028 |

[13] | Hwang, T.W., Uniqueness of limit cycles of the predator – prey system with beddington – deangelis functional response, J. math. anal. appl., 281, 395-401, (2003) · Zbl 1033.34052 |

[14] | Liu, Zhihua; Yuan, Rong, Stability and bifurcation in a delayed predator – prey system with beddington – deangelis functional response, J. math. anal. appl., 296, 521-537, (2004) · Zbl 1051.34060 |

[15] | Cantrell, R.S.; Cosner, C., Effect of domain size on the persistence of population in a diffusive food chain model with deangelis – beddington functional response, Natural resour. model., 14, 335-367, (2001) · Zbl 1005.92035 |

[16] | Cantrell, R.S.; Cosner, C., On the dynamics of predator – prey models with the beddington – deangelis functional response, J. math. anal. appl., 257, 206-222, (2001) · Zbl 0991.34046 |

[17] | Dai, Binxiang; Zhang, Na; Zou, Jiezhong, Permance for the michaelis – menten type discrete three-species ratio-dependent food chain model with delay, J. math. anal. appl., 324, 728-738, (2006) · Zbl 1101.92048 |

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