Dynamic complexities in 2-dimensional discrete-time predator-prey systems with Allee effect in the prey.

*(English)*Zbl 1376.92055Summary: The Allee effect is incorporated into a predator-prey model with linear functional response. Compared with the predator-prey which only takes the crowding effect and predator partially dependent on prey into consideration, it is found that the Allee effect of the prey species would increase the extinction risk of both the prey and predator. Moreover, by using a center manifold theorem and bifurcation theory, it is shown that the model with Allee effect undergoes the flip bifurcation and Hopf bifurcation in the interior of \(\mathbb R_+^2\) with different Allee effect values. In the two bifurcations, we can come to the conclusion that different Allee effect will have different bifurcation value and the increasing of the Allee effect will increase the value of bifurcation, respectively.

##### MSC:

92D25 | Population dynamics (general) |

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\textit{J. Yan} et al., Discrete Dyn. Nat. Soc. 2016, Article ID 4275372, 14 p. (2016; Zbl 1376.92055)

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##### References:

[1] | Berryman, A. A., The origins and evolution of predator-prey theory, Ecology, 73, 5, 1530-1535, (1992) |

[2] | Dhar, J., A prey-predator model with diffusion and a supplementary resource for the prey in a two-patch environment, Mathematical Modelling and Analysis, 9, 1, 9-24, (2004) · Zbl 1071.92039 |

[3] | Dhar, J.; Jatav, K. S., Mathematical analysis of a delayed stage-structured predator-prey model with impulsive diffusion between two predators territories, Ecological Complexity, 16, 59-67, (2013) |

[4] | Dubey, B., A prey-predator model with a reserved area, Nonlinear Analysis: Modelling and Control, 12, 4, 479-494, (2007) · Zbl 1206.49043 |

[5] | Freedman, H., Deterministic Mathematical Models in Population Ecology, (1980), Edmonton, Canada: HIFR Consulting, Edmonton, Canada · Zbl 0448.92023 |

[6] | Jeschke, J. M.; Kopp, M.; Tollrian, R., Predator functional responses: discriminating between handling and digesting prey, Ecological Monographs, 72, 1, 95-112, (2002) |

[7] | Kooij, R. E.; Zegeling, A., A predator-prey model with Ivlev’s functional response, Journal of Mathematical Analysis and Applications, 198, 2, 473-489, (1996) · Zbl 0851.34030 |

[8] | Lotka, A. J. |

[9] | Ma, W.; Takeuchi, Y., Stability analysis on a predator-prey system with distributed delays, Journal of Computational and Applied Mathematics, 88, 1, 79-94, (1998) · Zbl 0897.34062 |

[10] | May, R. M., Stability and Complexity in Model Ecosystems, 6, (2001), Princeton University Press |

[11] | Sen, M.; Banerjee, M.; Morozov, A., Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect, Ecological Complexity, 11, 12-27, (2012) |

[12] | Sinha, S.; Misra, O. P.; Dhar, J., Modelling a predator-prey system with infected prey in polluted environment, Applied Mathematical Modelling, 34, 7, 1861-1872, (2010) · Zbl 1193.34071 |

[13] | Volterra, V., Fluctuations in the abundance of a species considered mathematically, Nature, 118, 2972, 558-560, (1926) · JFM 52.0453.03 |

[14] | Dhar, J.; Singh, H.; Bhatti, H. S., Discrete-time dynamics of a system with crowding effect and predator partially dependent on prey, Applied Mathematics and Computation, 252, 324-335, (2015) · Zbl 1338.92100 |

[15] | Allee, W., Animal Aggregations: A Study in General Sociology, (1931), Chicago, Ill, USA: University of Chicago Press, Chicago, Ill, USA |

[16] | Conway, E. D.; Smoller, J. A., Global analysis of a system of predator-prey equations, SIAM Journal on Applied Mathematics, 46, 4, 630-642, (1986) · Zbl 0608.92016 |

[17] | Bazykin, A. D., Nonlinear Dynamics of Interacting Populations. Nonlinear Dynamics of Interacting Populations, World Scientific Series on Nonlinear Science Series A, 11, (1998), River Edge, NJ, USA: World Scientific, River Edge, NJ, USA |

[18] | González-Olivares, E.; Mena-Lorca, J.; Rojas-Palma, A.; Flores, J. D., Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey, Applied Mathematical Modelling, 35, 1, 366-381, (2011) · Zbl 1202.34079 |

[19] | Pal, P. J.; Mandal, P. K., Bifurcation analysis of a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and strong Allee effect, Mathematics and Computers in Simulation, 97, 123-146, (2014) |

[20] | Berec, L.; Angulo, E.; Courchamp, F., Multiple Allee effects and population management, Trends in Ecology and Evolution, 22, 4, 185-191, (2007) |

[21] | Boukal, D. S.; Berec, L., Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters, Journal of Theoretical Biology, 218, 3, 375-394, (2002) |

[22] | Boukal, D. S.; Sabelis, M. W.; Berec, L., How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses, Theoretical Population Biology, 72, 1, 136-147, (2007) · Zbl 1123.92034 |

[23] | Wang, G.; Liang, X.-G.; Wang, F.-Z., The competitive dynamics of populations subject to an Allee effect, Ecological Modelling, 124, 2-3, 183-192, (1999) |

[24] | González-Olivares, E.; González-Yañez, B.; Mena-Lorca, J.; Ramos-Jiliberto, R.; Mondaini, R., Modelling the Allee effect: are the different mathematical forms proposed equivalents?, Proceedings of the International Symposium on Mathematical and Computational Biology, E-Papers Serviços Editoriais Ltda |

[25] | Courchamp, F.; Clutton-Brock, T.; Grenfell, B., Inverse density dependence and the Allee effect, Trends in Ecology and Evolution, 14, 10, 405-410, (1999) |

[26] | Stephens, P. A.; Sutherland, W. J., Consequences of the Allee effect for behaviour, ecology and conservation, Trends in Ecology and Evolution, 14, 10, 401-405, (1999) |

[27] | Zu, J., Global qualitative analysis of a predator-prey system with Allee effect on the prey species, Mathematics and Computers in Simulation, 94, 33-54, (2013) |

[28] | Liu, X.; Xiao, D., Complex dynamic behaviors of a discrete-time predator-prey system, Chaos, Solitons & Fractals, 32, 1, 80-94, (2007) · Zbl 1130.92056 |

[29] | He, Z.; Lai, X., Bifurcation and chaotic behavior of a discrete-time predator-prey system, Nonlinear Analysis: Real World Applications, 12, 1, 403-417, (2011) · Zbl 1202.93038 |

[30] | Guckenheimer, J.; Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42, (1983), New York, NY, USA: Springer, New York, NY, USA · Zbl 0515.34001 |

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