We introduce a toy model of ecosystem assembly for which we are able to map out all assembly pathways generated by external invasions. The model allows us to display the whole phase space in the form of an assembly graph whose nodes are communities of species and whose directed links are transitions between them induced by invasions. We characterize the process as a finite Markov chain and prove that it exhibits a unique set of recurrent states (the end state of the process), which is therefore resistant to invasions. This also shows that the end state is independent of the assembly history. The model shares all features with standard assembly models reported in the literature, with the advantage that all observables can be computed in an exact manner.

We study the phase diagram of the double exchange model, with antiferromagnetic interactions, in a cubic lattice both at zero and finite temperature. There is a rich variety of magnetic phases, combined with regions where phase separation takes place. We identify phases, intrinsic to the cubic lattice, which are stable for realistic values of the interactions and dopings. Some of these phases break chiral symmetry, leading to unusual features.

If two species live on a single resource, the one with a slight advantage will out-compete the other: complete competitors cannot coexist. This is known as the competitive exclusion principle. If no extinction occurs, it is because evolutionary adaptation to slightly different niches takes place. Therefore, it is widely accepted that ecological communities are assembled by evolutionary differentiation and progressive adaptation of species to different niches. However, some ecologists have recently challenged this classic paradigm highlighting the importance of chance and stochasticity. Using a synthetic framework for community dynamics, here we show that, while deterministic descriptors predict coexistence, species similarity is limited in a more restrictive way in the presence of stochasticity. We analyse the stochastic extinction phenomenon, showing that extinction occurs as competitive overlap increases above a certain threshold well below its deterministic counterpart. We also prove that the extinction threshold cannot be ascribed only to demographic fluctuations around small population sizes. The more restrictive limit to species similarity is, therefore, a consequence of the complex interplay between competitive interactions and ecological drift. As a practical implication, we show that the existence of a stochastic limit to similarity has important consequences in the recovery of fragmented habitats.

The mesoscopic structure of complex networks has proven a powerful level of description to understand the linchpins of the system represented by the network. Nevertheless, the mapping of a series of relationships between elements, in terms of a graph, is sometimes not straightforward. Given that all the information we would extract using complex network tools depend on this initial graph, it is mandatory to preprocess the data to build it on in the most accurate manner. Here we propose a procedure to build a network, attending only to statistically significant relations between constituents. We use a paradigmatic example of word associations to show the development of our approach. Analyzing the modular structure of the obtained network we are able to disentangle categorical relations, disambiguating words with success that is comparable to the best algorithms designed to the same end. DOI: http://dx.doi.org/10.1103/PhysRevE.84.046108 Received 19 July 2011Revised 15 September 2011Published 19 October 2011©2011 American Physical Society