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

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

Drunken robber, tipsy cop: First passage times, mobile traps, and Hopf bifurcations

For a random walk on a confined one-dimensional domain, we consider mean first passage times (MFPT) in the presence of a mobile trap. The question we address is whether a mobile trap can improve capture times over a stationary trap. We consider two scenarios: a randomly moving trap and an oscillating trap. In both cases, we find that a stationary trap actually performs better (in terms of reducing expected capture time) than a very slowly moving trap; however, a trap moving sufficiently fast performs better than a stationary trap. We explicitly compute the thresholds that separate the two regimes. In addition, we find a surprising relation between the oscillating trap problem and a moving-sink problem that describes reduced dynamics of a single spike in a certain regime of the Gray-Scott model. Namely, the above-mentioned threshold corresponds precisely to a Hopf bifurcation that induces oscillatory motion in the location of the spike. We use this correspondence to prove the uniqueness of the Hopf bifurcation.Comment: 14 pages, 5 figure