J. C. Tzou

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 reg ime, and was previously analyzed in the context of ODE’s in [P.Mandel and T.Erneux, J.Stat.Phys 48(5-6) pp.1059-1070, 1987]. It was found that the instability would not be fully realized until the system had entered well into the unstab le 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 Giere r-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 s cenario, 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 syste m must pass well into the unstable regime before the onset of instability is fully observed, indicating delay. We also fin d 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 nonloca l eigenvalue problems (NLEP’s). The theory is confirmed by numerical solutions of the full PDE systems.

Asymptotic Analysis of First Passage Time Problems Inspired by Ecology

A hybrid asymptotic–numerical method is formulated and implemented to accurately calculate the mean first passage time (MFPT) for the expected time needed for a predator to locate small patches of prey in a 2-D landscape. In our analysis, the movement of the predator can have both a random and a directed component, where the diffusivity of the predator is isotropic but possibly spatially heterogeneous. Our singular perturbation methodology, which is based on the assumption that the ratio

Interaction of Turing and Hopf modes in the superdiffusive Brusselator model

\varepsilon

The stability of localized spikes for the 1-D Brusselator reaction-diffusion model

ε of the radius of a typical prey patch to that of the overall landscape is asymptotically small, leads to the derivation of an algebraic system that determines the MFPT in terms of parameters characterizing the shapes of the small prey patches together with a certain Green’s function, which in general must be computed numerically. The expected error in approximating the MFPT by our semi-analytical procedure is smaller than any power of

Mean First Passage Time for a Small Rotating Trap inside a Reflective Disk

{-1/\log \varepsilon }

Weakly Nonlinear Analysis of Vortex Formation in a Dissipative Variant of the Gross-Pitaevskii Equation

-1/logε, so that our approximation of the MFPT is still rather accurate at only moderately small prey patch radii. Overall, our hybrid approach has the advantage of eliminating the difficulty with resolving small spatial scales in a full numerical treatment of the partial differential equation (PDE). Similar semi-analytical methods are also developed and implemented to accurately calculate related quantities such as the variance of the mean first passage time (VMFPT) and the splitting probability. Results for the MFPT, the VMFPT, and splitting probability obtained from our hybrid methodology are validated with corresponding results computed from full numerical simulations of the underlying PDEs.