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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