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Hopf bifurcation in a reaction–diffusion equation with distributed delay and Dirichlet boundary condition.
- Source :
-
Journal of Differential Equations . Nov2017, Vol. 263 Issue 10, p6537-6575. 39p. - Publication Year :
- 2017
-
Abstract
- 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00220396
- Volume :
- 263
- Issue :
- 10
- Database :
- Academic Search Index
- Journal :
- Journal of Differential Equations
- Publication Type :
- Academic Journal
- Accession number :
- 125021854
- Full Text :
- https://doi.org/10.1016/j.jde.2017.07.024