参考文献/References:
[1] HUANG Jicai,LIU Sanhong,RUAN Shigui,et al.Bifurcations in a discrete predator-prey model with nonmonotonic functional respose[J].Journal of Mathematical Analysis and Applications,2018,464(1):201-230.DOI:10.1016/j.jmaa.2018.03.074.
[2] ZHONG Jiyu,YU Zhiheng.Qualitative properties and bifurcations of Mistro-Rodrigues-Petrovskii model[J].Nonlinear Dynamics,2018,91(4):2063-2075.DOI:10.1007/s11071-017-3932-0.
[3] DENNIS B.Allee effects: Population growth, critical density, and the change of extinction[J].Nature Resource Modeling,1989,3(4):481-538.DOI:10.1111/j.1939-7445.1989.tb00119.x.
[4] CHEN Xianwei,FU Xiangling,JING Zhujun.Dynamics in a discrete-time predator-prey system with Allee effect[J].Acta Mathematicae Applicatae Sinica, English Series,2013,29(1):143-164.DOI:10.1007/s10255-013-0207-5.
[5] WANG Wanxiong,ZHANG Yanbo,LIU Changzhong.Analysis of a discrete-time predator-prey system with Allee effect[J].Ecological Complexity,2011,8(1):81-85.DOI:10.1016/j.ecocom.2010.04.005.
[6] ZHANG Limin,ZHANG Chaofeng,HE Zhirong.Codimension-one and codimension-two bifurcations of a discrete predator-prey system with strong Allee effect[J].Mathematics and Computers in Simulation,2019,162:155-178.DOI:10.1016/j.matcom.2019.01.006.
[7] ZHANG Limin,XU Yike,LIAO Guangyuan.Codimension-two bifurcations and bifurcation controls in a discrete biological system with weak Allee effect[J].International Journal of Bifurcation and Chaos,2022,32(3):1-27.DOI:10.1142/S0218127422500365.
[8] PECORN N.Analysis of 1∶4 resonance in a monopoly model with memory[J].Chaos, Solitons and Fractals,2018,110:95-104.DOI:10.1016/j.chaos.2018.03.005.
[9] LI Bo,HE Zhimin.1∶2 and 1∶4 resonances in a two-dimensional discrete Hindmarsh-Rose model[J].Nonlinear Dynamics,2015,79(1):705-720.DOI:10.1007/s11071-014-1696-3.
[10] LUO Guangwei,ZHANG Yanlong,XIE Jianhua,et al.Periodic-impact motions and bifurcations of vibro-impact systems near 1∶4 strong resonance point[J].Communications in Nonlinear Science and Numerical Simulation,2008,13(5):1002-1014.DOI:10.1016/j.cnsns.2006.08.004.
[11] CHEN Qiaoling,TENG Zhidong,WANG Lei,et al.The existence of codimension-two bifurcation in a discrete SIS epidemic model with standard incidence[J].Nonlinear Dynamics,2013,71(1):55-73.DOI:10.1007/s11071-012-0641-6.
[12] CHAKRABORTY P,SARKAR S,GHOSH U.Stability and bifurcation analysis of a discrete prey-predator model with sigmoid functional response and Allee effect[J].Rendiconti del Circolo Matematico di Palermo Series 2,2021,70(1):253-273.DOI:10.1007/s12215-020-00495-5.
[13] KANGALGIL F,TOPSAKAL N,?ZTüRK N.Analyzing bifurcation, stability, and chaos control for a discrete-time prey-predator model with Allee effect[J].Turkish Journal of Mathematics,2022,46(6):2047-2068.DOI:10.55730/1300-0098.3253.
[14] CHENG Lifang,CAO Hongjun.Bifurcation analysis of a discrete-time ratio-dependent predator-prey model with Allee effect[J].Communications in Nonlinear Science and Numerical Simulation,2016,38(1):288-302.DOI:10.1016/j.cnsns.2016.02.038.
[15] CELIK C,DUMAN O.Allee effect in a discrete-time predator-prey system[J].Chaos, Solitons and Fractals,2009,40:1956-1962.DOI:10.1016/j.chaos.2007.09.077.
[16] 张芷芬,李承治,郑志明,等.向量场的分岔理论基础[M].北京:高等教育出版社,1997.
[17] KUZNETSOV Y A.Elements of applied bifurcation theory[M].2 ed.New York:Springer,1998.