Linking community ecology and biogeography: the role of biotic interactions and abiotic gradients in shaping the structure of ant communities.

Understanding what drives variation in species diversity in space and time and limits coexistence in local communities is a main focus of community ecology and biogeography. My doctoral work aims to document patterns of ant diversity and explore the possible ecological mechanisms leading to these patterns. Elucidating the processes by which communities assemble and species coexist might help explain spatial variation in species diversity. Using a combination of manipulative experiments, broad-scale surveys, behavioral assays and phylogenetic analyses, I examine which ecological processes account for the number of species coexisting in ant communities. Ants are found in most terrestrial habitats, where they are abundant, diverse and easy to sample (Agosti et al. 2000). Hölldobler and Wilson (1990) noted that competition was the hallmark of ant ecology, and we know that ant diversity varies along environmental gradients (Kusnezov 1957). Thus ants are an ideal taxon to examine the factors shaping the structure of ecological communities and how the determinants of community structure vary in space

Rigorous numerics for nonlinear operators with tridiagonal dominant linear part

We present a method designed for computing solutions of infinite dimensional non linear operators $f(x) = 0$ with a tridiagonal dominant linear part. We recast the operator equation into an equivalent Newton-like equation $x = T(x) = x - Af(x)$, where $A$ is an approximate inverse of the derivative $Df(\overline x)$ at an approximate solution $\overline x$. We present rigorous computer-assisted calculations showing that $T$ is a contraction near $\overline x$, thus yielding the existence of a solution. Since $Df(\overline x)$ does not have an asymptotically diagonal dominant structure, the computation of $A$ is not straightforward. This paper provides ideas for computing $A$, and proposes a new rigorous method for proving existence of solutions of nonlinear operators with tridiagonal dominant linear part.Comment: 27 pages, 3 figures, to be published in DCDS-A (Vol. 35, No. 10) October 2015 issu

Computation of maximal local (un)stable manifold patches by the parameterization method

In this work we develop some automatic procedures for computing high order polynomial expansions of local (un)stable manifolds for equilibria of differential equations. Our method incorporates validated truncation error bounds, and maximizes the size of the image of the polynomial approximation relative to some specified constraints. More precisely we use that the manifold computations depend heavily on the scalings of the eigenvectors: indeed we study the precise effects of these scalings on the estimates which determine the validated error bounds. This relationship between the eigenvector scalings and the error estimates plays a central role in our automatic procedures. In order to illustrate the utility of these methods we present several applications, including visualization of invariant manifolds in the Lorenz and FitzHugh-Nagumo systems and an automatic continuation scheme for (un)stable manifolds in a suspension bridge problem. In the present work we treat explicitly the case where the eigenvalues satisfy a certain non-resonance condition.Comment: Revised version, typos corrected, references adde

A method to rigorously enclose eigenpairs of complex interval matrices

summary:In this paper, a rigorous computational method to enclose eigenpairs of complex interval matrices is proposed. Each eigenpair x=(\lambda,\rv) is found by solving a nonlinear equation of the form $f(x)=0$ via a contraction argument. The set-up of the method relies on the notion of {\em radii polynomials}, which provide an efficient mean of determining a domain on which the contraction mapping theorem is applicable