We present three examples of delayed bifurcations for spike solutions of
reaction-diffusion systems. The delay effect results as the system passes
slowly from a stable to an unstable regime, and was previously analysed in the
context of ODE's in [P.Mandel, T.Erneux, J.Stat.Phys, 1987]. It was found that
the instability would not be fully realized until the system had entered well
into the unstable regime. The bifurcation is said to have been "delayed"
relative to the threshold value computed directly from a linear stability
analysis. In contrast, we analyze the delay effect in systems of PDE's. In
particular, for spike solutions of singularly perturbed generalized
Gierer-Meinhardt (GM) and Gray-Scott (GS) models, we analyze three examples of
delay resulting from slow passage into regimes of oscillatory and competition
instability. In the first example, for the GM model on the infinite real line,
we analyze the delay resulting from slowly tuning a control parameter through a
Hopf bifurcation. In the second example, we consider a Hopf bifurcation on a
finite one-dimensional domain. In this scenario, as opposed to the extrinsic
tuning of a system parameter through a bifurcation value, we analyze the delay
of a bifurcation triggered by slow intrinsic dynamics of the PDE system. In the
third example, we consider competition instabilities of the GS model triggered
by the extrinsic tuning of a feed rate parameter. In all cases, we find that
the system must pass well into the unstable regime before the onset of
instability is fully observed, indicating delay. We also find that delay has an
important effect on the eventual dynamics of the system in the unstable regime.
We give analytic predictions for the magnitude of the delays as obtained
through analysis of certain explicitly solvable nonlocal eigenvalue problems.
The theory is confirmed by numerical solutions of the full PDE systems.Comment: 31 pages, 20 figures, submitted to Physica D: Nonlinear Phenomen