具有比例时滞的离散Nicholson苍蝇模型的强共振

Strong resonance bifurcations of a discrete Nicholson′s blowflies model with proportional delay

种群模型是动力系统研究的一个重要分支,其中具有比例时滞的离散Nicholson苍蝇模型因其周期振动现象而广受关注.对该模型的稳定性和余维1分支情形,已有较多研究.本文旨在进一步讨论模型在余维2情形下的强共振分支现象.本文首先简要讨论了模型两个不动点的稳定性,然后利用规范标准型理论给出了模型在正不动点处分别产生1∶3和1∶4强共振的参数条件及其对应的规范标准型,最后展示了模型产生分支时的相图.研究强共振分支问题不仅有助于理解物种间相互作用对生物多样性和生态系统稳定性的影响,也有助于预测和管理生态系统的动态变化.

Population models have long been a focus of many researchers in the field of dynamical systems.Particularly,the discrete Nicholson’s blow flies model with proportional delay have attracted much attention due to its periodic oscillation.For the stability and co-dim 1 bifurcation of this model,many references exist.In this paper,we further discuss the co-dim 2 bifurcation and resonance of this model.First,the stability of two fixed points of model is briefly described.Then the parameter conditions for the model undergoing reso‐nances 1∶3 and 1∶4 at the positive fixed point and their normal form coefficients along with its scenario are dis‐cussed by using the normal form theory.Finally,bifurcation of the model is illustrated.Here we mention that the study of bifurcation and resonance can help us understand the effect of interaction among species on biodi‐versity and ecosystem stability and aid us predict and manage dynamic changes in ecosystems.