27 research outputs found
Early-Warning Signs for Pattern-Formation in Stochastic Partial Differential Equations
There have been significant recent advances in our understanding of the
potential use and limitations of early-warning signs for predicting drastic
changes, so called critical transitions or tipping points, in dynamical
systems. A focus of mathematical modeling and analysis has been on stochastic
ordinary differential equations, where generic statistical early-warning signs
can be identified near bifurcation-induced tipping points. In this paper, we
outline some basic steps to extend this theory to stochastic partial
differential equations with a focus on analytically characterizing basic
scaling laws for linear SPDEs and comparing the results to numerical
simulations of fully nonlinear problems. In particular, we study stochastic
versions of the Swift-Hohenberg and Ginzburg-Landau equations. We derive a
scaling law of the covariance operator in a regime where linearization is
expected to be a good approximation for the local fluctuations around
deterministic steady states. We compare these results to direct numerical
simulation, and study the influence of noise level, noise color, distance to
bifurcation and domain size on early-warning signs.Comment: Published in Communications in Nonlinear Science and Numerical
Simulation (2014
A topographic mechanism for arcing of dryland vegetation bands
Banded patterns consisting of alternating bare soil and dense vegetation have
been observed in water-limited ecosystems across the globe, often appearing
along gently sloped terrain with the stripes aligned transverse to the
elevation gradient. In many cases these vegetation bands are arced, with field
observations suggesting a link between the orientation of arcing relative to
the grade and the curvature of the underlying terrain. We modify the water
transport in the Klausmeier model of water-biomass interactions, originally
posed on a uniform hillslope, to qualitatively capture the influence of terrain
curvature on the vegetation patterns. Numerical simulations of this modified
model indicate that the vegetation bands change arcing-direction from
convex-downslope when growing on top of a ridge to convex-upslope when growing
in a valley. This behavior is consistent with observations from remote sensing
data that we present here. Model simulations show further that whether bands
grow on ridges, valleys, or both depends on the precipitation level. A survey
of three banded vegetation sites, each with a different aridity level,
indicates qualitatively similar behavior.Comment: 26 pages, 13 figures, 2 table
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An ensemble approach to the structure-function problem in microbial communities
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Genomic structure predicts metabolite dynamics in microbial communities
Data and code for "Genomic structure predicts metabolite dynamics in microbial communities," Cell (2022). https://doi.org/10.1016/j.cell.2021.12.03