Asymptotic Exit Location Distributions in the Stochastic Exit Problem

Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point $S$. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength $\epsilon$, the system state will eventually leave the domain of attraction $\Omega$ of $S$. We analyse the case when, as $\epsilon\to0$, the exit location on the boundary $\partial\Omega$ is increasingly concentrated near a saddle point $H$ of the deterministic dynamics. We show that the asymptotic form of the exit location distribution on $\partial\Omega$ is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter $\mu$, equal to the ratio $|\lambda_s(H)|/\lambda_u(H)$ of the stable and unstable eigenvalues of the linearized deterministic flow at $H$. If $\mu<1$ then the exit location distribution is generically asymptotic as $\epsilon\to0$ to a Weibull distribution with shape parameter $2/\mu$, on the $O(\epsilon^{\mu/2})$ length scale near $H$. If $\mu>1$ it is generically asymptotic to a distribution on the $O(\epsilon^{1/2})$ length scale, whose moments we compute. The asymmetry of the asymptotic exit location distribution is attributable to the generic presence of a `classically forbidden' region: a wedge-shaped subset of $\Omega$ with $H$ as vertex, which is reached from $S$, in the $\epsilon\to0$ limit, only via `bent' (non-smooth) fluctuational paths that first pass through the vicinity of $H$. We deduce from the presence of this forbidden region that the classical Eyring formula for the small-$\epsilon$ exponential asymptotics of the mean first exit time is generically inapplicable.Comment: This is a 72-page Postscript file, about 600K in length. Hardcopy requests to [email protected] or [email protected]

A Scaling Theory of Bifurcations in the Symmetric Weak-Noise Escape Problem

We consider the overdamped limit of two-dimensional double well systems perturbed by weak noise. In the weak noise limit the most probable fluctuational path leading from either point attractor to the separatrix (the most probable escape path, or MPEP) must terminate on the saddle between the two wells. However, as the parameters of a symmetric double well system are varied, a unique MPEP may bifurcate into two equally likely MPEP's. At the bifurcation point in parameter space, the activation kinetics of the system become non-Arrhenius. In this paper we quantify the non-Arrhenius behavior of a system at the bifurcation point, by using the Maslov-WKB method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix. The approximation is a formal asymptotic solution of the Smoluchowski equation. Our analysis relies on the development of a new scaling theory, which yields `critical exponents' describing weak-noise behavior near the saddle, at the bifurcation point.Comment: LaTeX, 60 pages, 24 Postscript figures. Uses epsf macros to include the figures. A file in `uufiles' format containing the figures is separately available at ftp://platinum.math.arizona.edu/pub/papers-rsm/paperF/figures.uu and a Postscript version of the whole paper (figures included) is available at ftp://platinum.math.arizona.edu/pub/papers-rsm/paperF/paperF.p

Lines in the sand : behaviour of self-organised vegetation patterns in dryland ecosystems

Vast, often populated, areas in dryland ecosystems face the dangers of desertification. Loosely speaking, desertification is the process in which a relatively dry region loses its vegetation - typically as an effect of climate change. As an important step in this process, the lack of resources forces the vegetation in these semi-arid areas to organise itself into large-scale spatial patterns. In this thesis, these patterns are studied using conceptual mathematical models, in which vegetation patterns present themselves as localised structures (for example pulses or fronts). These are analysed using mathematical techniques from (geometric singular) perturbation theory and via numerous numerical simulations. The study of these ecosystem models leads to new advances in both mathematics and ecology. NWO Mathematics of Planet EarthAnalysis and Stochastic