一类具双时滞的食饵-捕食系统的Hopf分支分析

 讨论了具有追捕时滞(捕食者的成熟时滞)τ₁、两种群生长时滞τ₂的双时滞食饵-捕食系统稳定性,确定各时滞取值范围对系统中食饵与捕食者数量的影响.首先,分别以追捕时滞(捕食者的成熟时滞)τ₁、两种群的生长时滞τ₂为参数,通过对系统线性化方程特征根的分布分析,找到特征根具严格负实部的时滞取值范围,得到系统平衡点的稳定性条件,确定了系统平衡点的线性稳定性区域.其次,讨论系统追捕时滞(捕食者的成熟时滞τ₁)在稳定区域内时,以两种群的生长时滞τ₂为参数时系统的稳定性,及系统Hopf分支的存在条件,利用中心流形理论和规范型方法计算了系统Hopf分支方向和分支周期解的稳定性,得到了系统参数的取值范围.

Hopf Bifurcation Analysis on a Predator-prey System with Double Time Delays

A kind of predator-prey system with double time delays i.e. the hunting delay (the time delay of the predator maturation) τ₁ and the growth time delay of the predator and the prey τ₂ is investigated. Stability analysis on the equilibrium point of the system is carried out, and the ranges of the time delays are determined. Thus, the impact of the predator and the prey is in-depth studied. First of all, the hunting delay (the time delay of the predator maturation) τ₁ and the growth time delay of the predator and the prey τ₂ are taken as parameters, respectively. The range of the roots with a strictly negative real part is found by analyzing the distribution of characteristic roots for the linearized equation. The condition to ensure the stability of the zero solution of the system and its stable domain are decided. Secondly, when the hunting delay τ₁ is in the stable region, the stability and the existence of Hopf bifurcation are given. Then the Hopf bifurcation is discussed by using the center manifold theory and normal form method introduced by Hassard and Kazarinoff. Both direction and stability of the Hopf bifurcation are considered, and the ranges for the system parameters are obtained.