一类带有强Allee效应捕食-食饵系统的Hopf分支

Hopf bifurcation in a predator-prey system with a strong Allee effect

研究了一类带有强Allee效应的HollingⅡ型功能反应函数的捕食-食饵动力系统.讨论了平衡点的存在性及稳定性,证明了在一定参数范围内存在Hopf分支,且由Hopf分支产生一个稳定的极限环,极限环随着同宿闭轨的产生而消失.还利用文献[13]的方法在一定的参数条件下把系统转化为一个Liénard-type系统,利用焦点的重数也可得到系统(3)存在一个稳定的极限环,同时利用数值模拟也证实了系统存在一个稳定的极限环.

A predator-prey model with a strong Allee effect in the prey was studied and the existence and stability of equilibrium and Hopf bifurcation was discussed. When the Hopf bifurcation occurred for an appropriate range of parameters, it was supercritical and disappeared due to a homoclinic bifurcation. This system was equivalent to the Liénard system in the range of parameters, and we gave a condition of equivalence to decide the existence and stability of the limit cycle. And numerical simulations were carried out to verify these results.