捕食者具有合作繁育的捕食者-食饵模型中的振荡

Oscillations in a Predatorprey Model with Cooperative Breeding of Predators

合作行为在动物中很常见.大量的研究表明,在捕食者食饵系统中合作捕食会引起复杂的动力学,例如平衡点共存和种群振荡.本文考虑一类具有合作繁育特征的捕食者食饵系统,并重点研究该系统的Hopf分岔及种群振荡问题.首先,讨论系统内部平衡点个数及其定性性质,证明在一定的参数条件下,平衡点E_(1)为中心型平衡点.其次分析在E_(1)处的Hopf分岔,并给出相应的分岔条件.最后,通过一个E_(1)为3阶细焦点的例子,研究系统复杂的振荡行为.由多个极限环的存在性得到:即使参数条件相同,周期和振幅也会随初始值的变化而变化,这反映了系统对初始值依赖的高度灵敏性.

The behavior of cooperation is common in animals.A lot of work showed that cooperative predation in predatorprey systems will induce complicated dynamics such as complicated coexistence of equilibria and population oscillations.In this paper,we consider a predatorprey system with cooperative breeding of predators,and focus on the oscillations related to the Hopf bifurcations of the model.Firstly,we discuss the number of internal equilibrium and the qualitative properties of internal equilibrium,and show that the equilibrium E_(1)is of center type under certain parameter conditions.Secondly,we analyze the Hopf bifurcation at E_(1),and give the corresponding bifurcation conditions.Finally,we give an example of a weak focus of order 3,and study the complex oscillation behaviors of the model through this example.Because of the coexistence of multiple limit cycles,even under the same parameter conditions,the periods and amplitudes vary with initial values.It also reflects the high sensitivity of solutions on initial values.