Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction–diffusion systems

• We present three examples of delayed bifurcations in partial differential equations.
• Bifurcations may be triggered by extrinsic tuning or intrinsic dynamics.
• Explicitly solvable nonlocal eigenvalue problems yield analytic results.
• Delay may play significant role in determining the eventual fate of the system.

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 analyzed in the context of ODE’s in Mandel and Erneux (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 to the study of Mandel and Erneux, we analyze the delay effect in systems of partial differential equations (PDE’s). In particular, for spike solutions of singularly perturbed generalized Gierer–Meinhardt and Gray–Scott models, we analyze three examples of delay resulting from slow passage into regimes of oscillatory and competition instability. In the first example, for the Gierer–Meinhardt 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 of the Gierer–Meinhardt model 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 triggered by the extrinsic tuning of a feed rate parameter. In all three 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 the analysis of certain explicitly solvable nonlocal eigenvalue problems (NLEP’s). The theory is confirmed by numerical solutions of the full PDE systems.