Differential Geometry for Model Independent Analysis of Images and Other Non-Euclidean Data: Recent Developments

This article provides an exposition of recent methodologies for nonparametric analysis of digital observations on images and other non-Euclidean objects. Fr\'echet means of distributions on metric spaces, such as manifolds and stratified spaces, have played an important role in this endeavor. Apart from theoretical issues of uniqueness of the Fr\'echet minimizer and the asymptotic distribution of the sample Fr\'echet mean under uniqueness, applications to image analysis are highlighted. In addition, nonparametric Bayes theory is brought to bear on the problems of density estimation and classification on manifolds

The Effect of Structural Complexity, Prey Density, and “Predator-Free Space” on Prey Survivorship at Created Oyster Reef Mesocosms

Interactions between predators and their prey are influenced by the habitat they occupy. Using created oyster (Crassostrea virginica) reef mesocosms, we conducted a series of laboratory experiments that created structure and manipulated complexity as well as prey density and “predator-free space” to examine the relationship between structural complexity and prey survivorship. Specifically, volume and spatial arrangement of oysters as well as prey density were manipulated, and the survivorship of prey (grass shrimp, Palaemonetes pugio) in the presence of a predator (wild red drum, Sciaenops ocellatus) was quantified. We found that the presence of structure increased prey survivorship, and that increasing complexity of this structure further increased survivorship, but only to a point. This agrees with the theory that structural complexity may influence predator-prey dynamics, but that a threshold exists with diminishing returns. These results held true even when prey density was scaled to structural complexity, or the amount of “predator-free space” was manipulated within our created reef mesocosms. The presence of structure and its complexity (oyster shell volume) were more important in facilitating prey survivorship than perceived refugia or density-dependent prey effects. A more accurate indicator of refugia might require “predator-free space” measures that also account for the available area within the structure itself (i.e., volume) and not just on the surface of a structure. Creating experiments that better mimic natural conditions and test a wider range of “predator-free space” are suggested to better understand the role of structural complexity in oyster reefs and other complex habitats