The Effects of Geography, Environment and Phylogeny on Community Assembly and Gene Flow Dynamics

It is increasingly evident that evolutionary processes play a role in how ecological communities are assembled. However the extend to which evolution influences how plants respond to spatial and environmental gradients and interact with each other is less clear. In this dissertation I leverage evolutionary tools and thinking to understand how space and environment affect community composition and patterns of gene flow in a unique system of Atlantic rainforest and restinga (sandy coastal plains) habitats in Southeastern Brazil.

In chapter one I investigate how space and environment affect the population genetic structure and gene flow of Aechmea nudicaulis, a bromeliad species that co-occurs in forest and restinga habitats. I genotyped seven microsatellite loci and sequenced one chloroplast DNA region for individuals collected in 7 pairs of forest / restinga sites. Bayesian genetic clustering analyses show that populations of A. nudicaulis are geographically structured in northern and southern populations, a pattern consistent with broader scale phylogeographic dynamics of the Atlantic rainforest. On the other hand, explicit migration models based on the coalescent estimate that inter-habitat gene flow is less common than gene flow between populations in the same habitat type, despite their geographic discontinuity. I conclude that there is evidence for repeated colonization of the restingas from forest populations even though the steep environmental gradient between habitats is a stronger barrier to gene flow than geographic distance.

In chapter two I use data on 2800 individual plants finely mapped in a restinga plot and on first-year survival of 500 seedlings to understand the roles of phylogeny, functional traits and abiotic conditions in the spatial structuring of that community. I demonstrate that phylogeny is a poor predictor of functional traits in and that convergence in these traits is pervasive. In general, the community is not phylogenetically structured, with at best 14% of the plots deviating significantly from the null model. The functional traits SLA, leaf dry matter content (LDMC), and maximum height also showed no clear pattern of spatial structuring. On the other hand, leaf area is strongly overdispersed across all spatial scales. Although leaf area overdispersion would be generally taken as evidence of competition, I argue that interpretation is probably misleading. Finally, I show that seedling survival is dramatically increased when they grow shaded by an adult individual, suggesting that seedlings are being facilitated. Phylogenetic distance to their adult neighbor has no influence on rates of survival though. Taken together, these results indicate that phylogeny has very limited influence on the fine scale assembly of restinga communities.

Application of Numerical Methods to Study Arrangement and Fracture of Lithium-Ion Microstructure

The focus of this work is to develop and employ numerical methods that provide characterization of granular microstructures, dynamic fragmentation of brittle materials, and dynamic fracture of three-dimensional bodies.

We first propose the fabric tensor formalism to describe the structure and evolution of lithium-ion electrode microstructure during the calendaring process. Fabric tensors are directional measures of particulate assemblies based on inter-particle connectivity, relating to the structural and transport properties of the electrode. Applying this technique to X-ray computed tomography of cathode microstructure, we show that fabric tensors capture the evolution of the inter-particle contact distribution and are therefore good measures for the internal state of and electronic transport within the electrode.

We then shift focus to the development and analysis of fracture models within finite element simulations. A difficult problem to characterize in the realm of fracture modeling is that of fragmentation, wherein brittle materials subjected to a uniform tensile loading break apart into a large number of smaller pieces. We explore the effect of numerical precision in the results of dynamic fragmentation simulations using the cohesive element approach on a one-dimensional domain. By introducing random and non-random field variations, we discern that round-off error plays a significant role in establishing a mesh-convergent solution for uniform fragmentation problems. Further, by using differing magnitudes of randomized material properties and mesh discretizations, we find that employing randomness can improve convergence behavior and provide a computational savings.

The Thick Level-Set model is implemented to describe brittle media undergoing dynamic fragmentation as an alternative to the cohesive element approach. This non-local damage model features a level-set function that defines the extent and severity of degradation and uses a length scale to limit the damage gradient. In terms of energy dissipated by fracture and mean fragment size, we find that the proposed model reproduces the rate-dependent observations of analytical approaches, cohesive element simulations, and experimental studies.

Lastly, the Thick Level-Set model is implemented in three dimensions to describe the dynamic failure of brittle media, such as the active material particles in the battery cathode during manufacturing. The proposed model matches expected behavior from physical experiments, analytical approaches, and numerical models, and mesh convergence is established. We find that the use of an asymmetrical damage model to represent tensile damage is important to producing the expected results for brittle fracture problems.

The impact of this work is that designers of lithium-ion battery components can employ the numerical methods presented herein to analyze the evolving electrode microstructure during manufacturing, operational, and extraordinary loadings. This allows for enhanced designs and manufacturing methods that advance the state of battery technology. Further, these numerical tools have applicability in a broad range of fields, from geotechnical analysis to ice-sheet modeling to armor design to hydraulic fracturing.

Convergent differential regulation of parvalbumin in the brains of vocal learners.

Spoken language and learned song are complex communication behaviors found in only a few species, including humans and three groups of distantly related birds--songbirds, parrots, and hummingbirds. Despite their large phylogenetic distances, these vocal learners show convergent behaviors and associated brain pathways for vocal communication. However, it is not clear whether this behavioral and anatomical convergence is associated with molecular convergence. Here we used oligo microarrays to screen for genes differentially regulated in brain nuclei necessary for producing learned vocalizations relative to adjacent brain areas that control other behaviors in avian vocal learners versus vocal non-learners. A top candidate gene in our screen was a calcium-binding protein, parvalbumin (PV). In situ hybridization verification revealed that PV was expressed significantly higher throughout the song motor pathway, including brainstem vocal motor neurons relative to the surrounding brain regions of all distantly related avian vocal learners. This differential expression was specific to PV and vocal learners, as it was not found in avian vocal non-learners nor for control genes in learners and non-learners. Similar to the vocal learning birds, higher PV up-regulation was found in the brainstem tongue motor neurons used for speech production in humans relative to a non-human primate, macaques. These results suggest repeated convergent evolution of differential PV up-regulation in the brains of vocal learners separated by more than 65-300 million years from a common ancestor and that the specialized behaviors of learned song and speech may require extra calcium buffering and signaling.

Stochastic switching in infinite dimensions with applications to random parabolic PDE

© 2015 Society for Industrial and Applied Mathematics.We consider parabolic PDEs with randomly switching boundary conditions. In order to analyze these random PDEs, we consider more general stochastic hybrid systems and prove convergence to, and properties of, a stationary distribution. Applying these general results to the heat equation with randomly switching boundary conditions, we find explicit formulae for various statistics of the solution and obtain almost sure results about its regularity and structure. These results are of particular interest for biological applications as well as for their significant departure from behavior seen in PDEs forced by disparate Gaussian noise. Our general results also have applications to other types of stochastic hybrid systems, such as ODEs with randomly switching right-hand sides.

Bayesian Inference in Large-scale Problems

Many modern applications fall into the category of "large-scale" statistical problems, in which both the number of observations n and the number of features or parameters p may be large. Many existing methods focus on point estimation, despite the continued relevance of uncertainty quantification in the sciences, where the number of parameters to estimate often exceeds the sample size, despite huge increases in the value of n typically seen in many fields. Thus, the tendency in some areas of industry to dispense with traditional statistical analysis on the basis that "n=all" is of little relevance outside of certain narrow applications. The main result of the Big Data revolution in most fields has instead been to make computation much harder without reducing the importance of uncertainty quantification. Bayesian methods excel at uncertainty quantification, but often scale poorly relative to alternatives. This conflict between the statistical advantages of Bayesian procedures and their substantial computational disadvantages is perhaps the greatest challenge facing modern Bayesian statistics, and is the primary motivation for the work presented here.

Two general strategies for scaling Bayesian inference are considered. The first is the development of methods that lend themselves to faster computation, and the second is design and characterization of computational algorithms that scale better in n or p. In the first instance, the focus is on joint inference outside of the standard problem of multivariate continuous data that has been a major focus of previous theoretical work in this area. In the second area, we pursue strategies for improving the speed of Markov chain Monte Carlo algorithms, and characterizing their performance in large-scale settings. Throughout, the focus is on rigorous theoretical evaluation combined with empirical demonstrations of performance and concordance with the theory.

One topic we consider is modeling the joint distribution of multivariate categorical data, often summarized in a contingency table. Contingency table analysis routinely relies on log-linear models, with latent structure analysis providing a common alternative. Latent structure models lead to a reduced rank tensor factorization of the probability mass function for multivariate categorical data, while log-linear models achieve dimensionality reduction through sparsity. Little is known about the relationship between these notions of dimensionality reduction in the two paradigms. In Chapter 2, we derive several results relating the support of a log-linear model to nonnegative ranks of the associated probability tensor. Motivated by these findings, we propose a new collapsed Tucker class of tensor decompositions, which bridge existing PARAFAC and Tucker decompositions, providing a more flexible framework for parsimoniously characterizing multivariate categorical data. Taking a Bayesian approach to inference, we illustrate empirical advantages of the new decompositions.

Latent class models for the joint distribution of multivariate categorical, such as the PARAFAC decomposition, data play an important role in the analysis of population structure. In this context, the number of latent classes is interpreted as the number of genetically distinct subpopulations of an organism, an important factor in the analysis of evolutionary processes and conservation status. Existing methods focus on point estimates of the number of subpopulations, and lack robust uncertainty quantification. Moreover, whether the number of latent classes in these models is even an identified parameter is an open question. In Chapter 3, we show that when the model is properly specified, the correct number of subpopulations can be recovered almost surely. We then propose an alternative method for estimating the number of latent subpopulations that provides good quantification of uncertainty, and provide a simple procedure for verifying that the proposed method is consistent for the number of subpopulations. The performance of the model in estimating the number of subpopulations and other common population structure inference problems is assessed in simulations and a real data application.

In contingency table analysis, sparse data is frequently encountered for even modest numbers of variables, resulting in non-existence of maximum likelihood estimates. A common solution is to obtain regularized estimates of the parameters of a log-linear model. Bayesian methods provide a coherent approach to regularization, but are often computationally intensive. Conjugate priors ease computational demands, but the conjugate Diaconis--Ylvisaker priors for the parameters of log-linear models do not give rise to closed form credible regions, complicating posterior inference. In Chapter 4 we derive the optimal Gaussian approximation to the posterior for log-linear models with Diaconis--Ylvisaker priors, and provide convergence rate and finite-sample bounds for the Kullback-Leibler divergence between the exact posterior and the optimal Gaussian approximation. We demonstrate empirically in simulations and a real data application that the approximation is highly accurate, even in relatively small samples. The proposed approximation provides a computationally scalable and principled approach to regularized estimation and approximate Bayesian inference for log-linear models.

Another challenging and somewhat non-standard joint modeling problem is inference on tail dependence in stochastic processes. In applications where extreme dependence is of interest, data are almost always time-indexed. Existing methods for inference and modeling in this setting often cluster extreme events or choose window sizes with the goal of preserving temporal information. In Chapter 5, we propose an alternative paradigm for inference on tail dependence in stochastic processes with arbitrary temporal dependence structure in the extremes, based on the idea that the information on strength of tail dependence and the temporal structure in this dependence are both encoded in waiting times between exceedances of high thresholds. We construct a class of time-indexed stochastic processes with tail dependence obtained by endowing the support points in de Haan's spectral representation of max-stable processes with velocities and lifetimes. We extend Smith's model to these max-stable velocity processes and obtain the distribution of waiting times between extreme events at multiple locations. Motivated by this result, a new definition of tail dependence is proposed that is a function of the distribution of waiting times between threshold exceedances, and an inferential framework is constructed for estimating the strength of extremal dependence and quantifying uncertainty in this paradigm. The method is applied to climatological, financial, and electrophysiology data.

The remainder of this thesis focuses on posterior computation by Markov chain Monte Carlo. The Markov Chain Monte Carlo method is the dominant paradigm for posterior computation in Bayesian analysis. It has long been common to control computation time by making approximations to the Markov transition kernel. Comparatively little attention has been paid to convergence and estimation error in these approximating Markov Chains. In Chapter 6, we propose a framework for assessing when to use approximations in MCMC algorithms, and how much error in the transition kernel should be tolerated to obtain optimal estimation performance with respect to a specified loss function and computational budget. The results require only ergodicity of the exact kernel and control of the kernel approximation accuracy. The theoretical framework is applied to approximations based on random subsets of data, low-rank approximations of Gaussian processes, and a novel approximating Markov chain for discrete mixture models.

Data augmentation Gibbs samplers are arguably the most popular class of algorithm for approximately sampling from the posterior distribution for the parameters of generalized linear models. The truncated Normal and Polya-Gamma data augmentation samplers are standard examples for probit and logit links, respectively. Motivated by an important problem in quantitative advertising, in Chapter 7 we consider the application of these algorithms to modeling rare events. We show that when the sample size is large but the observed number of successes is small, these data augmentation samplers mix very slowly, with a spectral gap that converges to zero at a rate at least proportional to the reciprocal of the square root of the sample size up to a log factor. In simulation studies, moderate sample sizes result in high autocorrelations and small effective sample sizes. Similar empirical results are observed for related data augmentation samplers for multinomial logit and probit models. When applied to a real quantitative advertising dataset, the data augmentation samplers mix very poorly. Conversely, Hamiltonian Monte Carlo and a type of independence chain Metropolis algorithm show good mixing on the same dataset.

Convergence of numerical time-averaging and stationary measures via Poisson equations

Numerical approximation of the long time behavior of a stochastic di.erential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantages of this approach are its simplicity and universality. It works equally well for a range of explicit and implicit schemes, including those with simple simulation of random variables, and for hypoelliptic SDEs. To simplify the exposition, we consider only the case where the state space of the SDE is a torus, and we study only smooth test functions. However, we anticipate that the approach can be applied more widely. An analogy between our approach and Stein's method is indicated. Some practical implications of the results are discussed. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

Accelerating self-consistent field convergence with the augmented Roothaan-Hall energy function.

Based on Pulay's direct inversion iterative subspace (DIIS) approach, we present a method to accelerate self-consistent field (SCF) convergence. In this method, the quadratic augmented Roothaan-Hall (ARH) energy function, proposed recently by Høst and co-workers [J. Chem. Phys. 129, 124106 (2008)], is used as the object of minimization for obtaining the linear coefficients of Fock matrices within DIIS. This differs from the traditional DIIS of Pulay, which uses an object function derived from the commutator of the density and Fock matrices. Our results show that the present algorithm, abbreviated ADIIS, is more robust and efficient than the energy-DIIS (EDIIS) approach. In particular, several examples demonstrate that the combination of ADIIS and DIIS ("ADIIS+DIIS") is highly reliable and efficient in accelerating SCF convergence.

Three-dimensional dispersive metallic photonic crystals with a bandgap and a high cutoff frequency.

The goal of this work is to analyze three-dimensional dispersive metallic photonic crystals (PCs) and to find a structure that can provide a bandgap and a high cutoff frequency. The determination of the band structure of a PC with dispersive materials is an expensive nonlinear eigenvalue problem; in this work we propose a rational-polynomial method to convert such a nonlinear eigenvalue problem into a linear eigenvalue problem. The spectral element method is extended to rapidly calculate the band structure of three-dimensional PCs consisting of realistic dispersive materials modeled by Drude and Drude-Lorentz models. Exponential convergence is observed in the numerical experiments. Numerical results show that, at the low frequency limit, metallic materials are similar to a perfect electric conductor, where the simulation results tend to be the same as perfect electric conductor PCs. Band structures of the scaffold structure and semi-woodpile structure metallic PCs are investigated. It is found that band structures of semi-woodpile PCs have a very high cutoff frequency as well as a bandgap between the lowest two bands and the higher bands.

Decoding an olfactory mechanism of kin recognition and inbreeding avoidance in a primate.

BACKGROUND: Like other vertebrates, primates recognize their relatives, primarily to minimize inbreeding, but also to facilitate nepotism. Although associative, social learning is typically credited for discrimination of familiar kin, discrimination of unfamiliar kin remains unexplained. As sex-biased dispersal in long-lived species cannot consistently prevent encounters between unfamiliar kin, inbreeding remains a threat and mechanisms to avoid it beg explanation. Using a molecular approach that combined analyses of biochemical and microsatellite markers in 17 female and 19 male ring-tailed lemurs (Lemur catta), we describe odor-gene covariance to establish the feasibility of olfactory-mediated kin recognition. RESULTS: Despite derivation from different genital glands, labial and scrotal secretions shared about 170 of their respective 338 and 203 semiochemicals. In addition, these semiochemicals encoded information about genetic relatedness within and between the sexes. Although the sexes showed opposite seasonal patterns in signal complexity, the odor profiles of related individuals (whether same-sex or mixed-sex dyads) converged most strongly in the competitive breeding season. Thus, a strong, mutual olfactory signal of genetic relatedness appeared specifically when such information would be crucial for preventing inbreeding. That weaker signals of genetic relatedness might exist year round could provide a mechanism to explain nepotism between unfamiliar kin. CONCLUSION: We suggest that signal convergence between the sexes may reflect strong selective pressures on kin recognition, whereas signal convergence within the sexes may arise as its by-product or function independently to prevent competition between unfamiliar relatives. The link between an individual's genome and its olfactory signals could be mediated by biosynthetic pathways producing polymorphic semiochemicals or by carrier proteins modifying the individual bouquet of olfactory cues. In conclusion, we unveil a possible olfactory mechanism of kin recognition that has specific relevance to understanding inbreeding avoidance and nepotistic behavior observed in free-ranging primates, and broader relevance to understanding the mechanisms of vertebrate olfactory communication.

Love is in the air: Sociality and pair bondedness influence sifaka reproductive signalling

© 2013 The Association for the Study of Animal Behaviour.Social complexity, often estimated by group size, is seen as driving the complexity of vocal signals, but its relation to olfactory signals, which arguably arose to function in nonsocial realms, remains underappreciated. That olfactory signals also may mediate within-group interaction, vary with social complexity and promote social cohesion underscores a potentially crucial link with sociality. To examine that link, we integrated chemical and behavioural analyses to ask whether olfactory signals facilitate reproductive coordination in a strepsirrhine primate, the Coquerel's sifaka, Propithecus coquereli. Belonging to a clade comprising primarily solitary, nocturnal species, the diurnal, group-living sifaka represents an interesting test case. Convergent with diurnal, group-living lemurids, sifakas expressed chemically rich scent signals, consistent with the social complexity hypothesis for communication. These signals minimally encoded the sex of the signaller and varied with female reproductive state. Likewise, sex and female fertility were reflected in within-group scent investigation, scent marking and overmarking. We further asked whether, within breeding pairs, the stability or quality of the pair's bond influences the composition of glandular signals and patterns of investigatory or scent-marking behaviour. Indeed, reproductively successful pairs tended to show greater similarity in their scent signals than did reproductively unsuccessful pairs, potentially through chemical convergence. Moreover, scent marking was temporally coordinated within breeding pairs and was influenced by past reproductive success. That olfactory signalling reflects social bondedness or reproductive history lends support to recent suggestions that the quality of relationships may be a more valuable proxy than group size for estimating social complexity. We suggest that olfactory signalling in sifakas is more complex than previously recognized and, as in other socially integrated species, can be a crucial mechanism for promoting group cohesion and maintaining social bonds. Thus, the evolution of sociality may well be reflected in the complexity of olfactory signalling.