Feasibility and coexistence of large ecological communities

The role of species interactions in controlling the interplay between the stability of ecosystems and their biodiversity is still not well understood. The ability of ecological communities to recover after small perturbations of the species abundances (local asymptotic stability) has been well studied, whereas the likelihood of a community to persist when the conditions change (structural stability) has received much less attention. Our goal is to understand the effects of diversity, interaction strengths and ecological network structure on the volume of parameter space leading to feasible equilibria. We develop a geometrical framework to study the range of conditions necessary for feasible coexistence. We show that feasibility is determined by few quantities describing the interactions, yielding a nontrivial complexity–feasibility relationship. Analysing more than 100 empirical networks, we show that the range of coexistence conditions in mutualistic systems can be analytically predicted. Finally, we characterize the geometric shape of the feasibility domain, thereby identifying the direction of perturbations that are more likely to cause extinctions

A New Monte Carlo Based Algorithm for the Gaussian Process Classification Problem

Gaussian process is a very promising novel technology that has been applied to both the regression problem and the classification problem. While for the regression problem it yields simple exact solutions, this is not the case for the classification problem, because we encounter intractable integrals. In this paper we develop a new derivation that transforms the problem into that of evaluating the ratio of multivariate Gaussian orthant integrals. Moreover, we develop a new Monte Carlo procedure that evaluates these integrals. It is based on some aspects of bootstrap sampling and acceptancerejection. The proposed approach has beneficial properties compared to the existing Markov Chain Monte Carlo approach, such as simplicity, reliability, and speed

A structural approach for understanding multispecies coexistence

Although observations of species-rich communities have long served as a primary motivation for research on the coexistence of competitors, the majority of our empirical and theoretical understanding comes from two-species systems. How much of the coexistence observed in species rich communities results from indirect effects among competitors that only emerge in diverse systems remains poorly understood. Resolving this issue requires simple, scalable, and intuitive metrics for quantifying the conditions for coexistence in multispecies systems, and how these conditions differ from those expected based solely on pairwise interactions. To achieve these aims, we develop a structural approach for studying the set of parameter values compatible with n-species coexistence given the geometric constraints imposed by the the matrix of competition coefficients. We derive novel mathematical metrics analogous to stabilizing niche differences and fitness differences that measure the range of conditions compatible with multispecies coexistence, incorporating the effects of indirect interactions emerging in diverse systems. We show how our measures can be used to quantify the extent to which the conditions for coexistence in multispecies systems differ from those that allow pairwise coexistence, and apply the method to a field system of annual plants. We conclude by presenting new challenges and empirical opportunities emerging from our structural metrics of multispecies coexistence

Solid angle measure of polyhedral cones

This paper addresses the computation of normalized solid angle measure of polyhedral cones. This is well understood in dimensions two and three. For higher dimensions, assuming that a positive-definite criterion is met, the measure can be computed via a multivariable hypergeometric series. We present two decompositions of full-dimensional simplicial cones into finite families of cones satisfying the positive-definite criterion, enabling the use of the hypergeometric series to compute the solid angle measure of any polyhedral cone. Additionally, our second decomposition method yields cones with a special tridiagonal structure, reducing the number of required coordinates for the hypergeometric series formula. Furthermore, we investigate the convergence of the hypergeometric series for this case. Our findings provide a powerful tool for computing solid angle measures in high-dimensional spaces