密度制约且具有时滞捕食-被捕食模型的Hopf分支

Hopf Bifurcation of Delayed Density-Dependent Predator-Prey Model

捕食者和被捕食者都具有密度制约的捕食-被捕食模型更符合实际的生物环境,也更有一定的现实意义,所以该文考虑了密度制约且具有Beddington-DeAngelis功能反应函数的时滞捕食-被捕食系统.首先,给出系统的稳定性变换为后面讨论分支周期解做好准备.其次,由中心流形定理和规范型理论,得到Hopf分支关于分支方向、稳定性以及周期情况的特性指标.最后,在数值模拟这部分对系统稳定性变换和系统的Hopf分支进行了验证.

In this paper, we investigate stability and Hopf bifurcation of a delayed density-dependent predator-prey system with Beddington-DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered such that the studied predator-prey system conforms to the realistically biological environment. Firstly, stability transformation of the system was given to prepare for discussion of bifurcating periodic solution. Secondly, we discussed properties of Hopf bifurcation about bifurcating direction, stability and period by center manifold theorem and normal form theory. Finally, an example with numerical simulations is given to illustrate stability transformation and Hopf bifurcation of the system.