Hopf bifurcation in a reaction-diffusion equation with distributed delay and Dirichlet boundary condition
Hopf bifurcation in a reaction-diffusion equation with distributed delay and Dirichlet boundary condition
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具有分布延迟和狄利克雷边界条件的反应扩散方程中的 Hopf 分岔
DOI：
10.1016/j.jde.2017.07.024
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发表时间：
2017
影响因子：
2.4
通讯作者：
Song Yongli
中科院分区：
文献类型：
--
作者：
Shi Qingyan;Shi Junping;Song Yongli
研究了一类具有Dirichlet边界条件和分布时滞的一般标量反应扩散方程正平衡态的稳定性和Hopf分支.时间延迟遵循伽马分布函数。通过分析相应的特征值问题，严格证明了当形状参数n≥ 1时，系统将发生Hopf分支，当n= 0时，系统的定态总是稳定的.通过计算中心流形上的规范形，在一般情况下，也可以确定系统的Hopf分支方向和周期轨道的稳定性。结果表明，与离散时滞情形相比，Hopf分支的临界时滞数是有限的，且随n的增加而增加，第一个Hopf分支值随n的增加而减小.从种群生物学和数值模拟的例子来说明理论结果。
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 in n, 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.