The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology

Dynamical analysis of a logistic equation with spatio-temporal delay

“…The results in the literatures [19,17,18] show that changes of delay for Eq. (1.2) do lead to the occurrence of Hopf bifurcation when the delay crosses through a sequence of critical values.…”

“…For example, for the Neumann boundary value problem, (1.2) has been considered in [17,18] and they considered the stability and related Hopf bifurcation from the homogeneous equilibrium. Busenberg and Huang [19] studied the Hopf bifurcation of (1.2) and Dirichlet boundary condition proposed by Green and Stech [20].…”

Destabilization of periodic solutions arising in delay-diffusion systems in several space dimensions

“…As for stability of the spatially homogeneous periodic solution to (E3), Yoshida [7] has proved that the bifurcating periodic solution near the bifurcation point, a = 7r/2, is stable. However, ir has not been made clear in [7] how the stability region of the bifurcation parameter, a, depends on other factors such as the diffusion constant, d, and the shape of the domain, t2.…”

“…However, ir has not been made clear in [7] how the stability region of the bifurcation parameter, a, depends on other factors such as the diffusion constant, d, and the shape of the domain, t2.…”

“…As mentioned in the Introduction, Yoshida [7] has shown for some specific equation that the spatially homogeneous periodic solution p(. ; e) is stable in the right above sense near the bifurcation point.…”