Assessing the relative importance of intransitive competition networks in nature has been difficult because it requires a large number of pairwise competition experiments linked to observed field abundances of interacting species. Here we introduce metrics and statistical tests for evaluating the contribution of intransitivity to community structure using two kinds of data: competition matrices derived from the outcomes of pairwise experimental studies (C matrices) and species abundance matrices. We use C matrices to develop patch transition matrices (P) that predict community structure in a simple Markov chain model. We propose a randomization test to evaluate the degree of intransitivity from these P matrices in combination with empirical or simulated C matrices. Benchmark tests revealed that the methods could correctly detect intransitive competition networks, even in the absence of direct measures of pairwise competitive strength. These tests represent the first tools for estimating the degree of intransitivity in competitive networks from observational datasets. They can be applied to both spatio-temporal data sampled in homogeneous environments or across environmental gradients, and to experimental measures of pairwise interactions. To illustrate the methods, we analyzed empirical data matrices on the colonization of slug carrion by necrophagous flies and their parasitoids.
Binary presence-absence matrices (rows species, columns sites) are often used to quantify patterns of species co-occurrence, and to infer possible biotic interactions from these patterns. Previous classifications of co-occurrence patterns as nested, segregated, or modular have led to contradictory results and conclusions. These analyses usually do not incorporate the functional traits of the species or the environmental characteristics of the sites, even though the outcomes of species interactions often depend on trait expression and site quality. Here we address this shortcoming by developing a method that incorporates realized functional and environmental niches, and relates them to species co-occurrence patterns. These niches are defined from n-dimensional ellipsoids, and calculated from the n eigenvectors and eigenvalues of the variancecovariance matrix of measured environmental or trait variables. Average niche overlap among species and the spatial distribution of niches define a triangle plot with vertices of species segregation (low niche overlap), nestedness (high niche overlap), and modular co-occurrence (clusters of overlapping niches). Applying this framework to temperate understorey plant communities in southwest Poland, we found a consistent modular structure of species occurrences, a pattern not detected by conventional presence-absence analysis. These results suggest that, in our case study, habitat filtering is the most important process structuring understorey plant communities. Furthermore, they demonstrate how incorporating trait and environmental data into co-occurrence analysis improves pattern detection and provides a stronger theoretical framework for understanding community structure.
Abstract. In this paper we construct an infinite dimensional (extraordinary) cohomology theory and a Morse theory corresponding to it. These theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals (by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension). Several applications are given to Hamiltonian systems, the onedimensional wave equation (of vibrating string type) and systems of elliptic partial differential equations.
In the first part of the paper we provide a construction of an abstract homotopy invariant detecting zeros of maps of the formon an open subset U of a neighborhood retract M being invariant with respect to the resolvents of A. The construction is performed under the assumption that resolvents of A are completely continuous. In the second part we derive index formulae for isolated zeros and apply them to show the existence of nontrivial positive steady state solutions for a class of nonlinear reaction-diffusion equations and equations with one-dimensional p-Laplacian with possibly non-positive perturbations as well as some controlled Neumann-like problems.
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