一类具有扩散的捕食系统的Hopf分岔分析

The Hopf Bifurcation Analysis of a Reaction Diffusion Predator-prey System

动力系统有着其复杂的一面,分岔是一种常见的重要的非线性现象,并与其他非线性现象(如混沌、突变、分形等)密切相关;不同类型的捕食者-食饵模型已经得到了广泛的研究和长足的发展,因此无论是从生物学家还是数学家来说考虑二者的动力学关系和分岔现象都是十分有意义的。本文研究了一类具有扩散的捕食系统,在Neumann边界条件下考虑扩散捕食系统的稳定性和H opf分岔,通过中心流形定理和规范型理论,得到了在一定的条件下系统有一族实得周期解,当参数足够小时,在某一点发生分岔;当线性算子的所有特征值都有非零实部时,周期解是稳定的,否则是不稳定的。

As power system is complicated, bifurcation is a common but important non-linear phenomena closely related to some other nonlinear phenomenon, such as the chaos, mutation, fractals, etc.. Different types of prey-predator model have got extensive research and rapid development, so it＇s meaningful to study their dynamic relation and bifurcation phenomenon in either biological way or mathematical way. This paper has studied a reaction diffusion predator-prey system where stability and Hopf bifurcation are considered under the Neumann boundary conditions. Through the center manifold theorem and the normal form theory, a conclusion can be reached that the system has a series of periodic solutions under the given conditions. When the parameters are small enough, the bifurcation takes place at some point; when all the characteristic values of the linear operators have the nonzero real part, the periodic solution is stable, otherwise it is unstable.