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

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]