具有时滞和线性收获项的偏利合作系统的Hopf分支

Hopf Bifurcation of a Commensalism System with Time Delay and Linear Harvesting Effect

以时滞τ作为分支参数,研究了具有时滞和线性收获项的偏利合作系统的动力学行为.首先,利用Cooke等人给出的关于超越函数的零点分布定理,分析了系统唯一正平衡点的稳定性及局部Hopf分支的存在性,找到了使系统产生局部Hopf分支的分支值;其次,运用Hassard的规范型理论和中心流形定理,得到了确定Hopf分支方向与分支周期解稳定性的计算公式;最后,使用Matlab软件进行数值模拟,验证了理论分析结果的可行性.

With time delay τ as a bifurcation parameter,the dynamic behaviors in commensalism system with time delay and linear harvesting effect were studied.Firstly,based on the zero point distribution theorem of the transcendental function given by Cooke et al,the stability of the system positive equilibrium point and the existence of the local Hopf bifurcation were researched.The local Hopf bifurcation value of the system was obtained.Secondly,the direction and the stability of the Hopf bifurcating periodic solution were determined by Hassard's normal form theory and the center manifold theorem.Finally,we used the Matlab to carry out some numerical simulations to support the theoretical findings.