Hopf bifurcation in a reaction-diffusion equation with distributed delay and Dirichlet boundary condition

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:263

Article

Shi, Qingyan¹  Shi, Junping²  Song, Yongli³ 

[1] Tongji Univ, Dept Math, Shanghai 200092, Peoples R China

[2] Coll William & Mary, Dept Math, Williamsburg, VA 23187 USA

[3] Hangzhou Normal Univ, Dept Math, Hangzhou 311121, Zhejiang, Peoples R China

关键词: Hopf bifurcation;    Distributed delay;    Dirichlet boundary condition;    Reaction-diffusion equation;    Normal form;   

DOI  :  10.1016/j.jde.2017.07.024

来源: Elsevier

【 摘 要 】

The stability and Hopf bifurcation of the positive steady state to a general scalar reaction diffusion equation with distributed delay and Dirichlet boundary condition are investigated in this paper. The time delay follows a Gamma distribution function. Through analyzing the corresponding eigenvalue problems, we rigorously show that Hopf bifurcations will occur when the shape parameter n >= 1, and the steady state is always stable when n = 0. By computing normal form on the center manifold, the direction of Hopf bifurcation and the stability of the periodic orbits can also be determined under a general setting. Our results show that the number of critical values of delay for Hopf bifurcation is finite and increasing inn, which is significantly different from the discrete delay case, and the first Hopf bifurcation value is decreasing in n. Examples from population biology and numerical simulations are used to illustrate the theoretical results. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free