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 phenomenon of length-scale competition in soliton-bearing equations perturbed by spatially dependent terms [A. Sánchez and A. R. Bishop, SIAM Rev. 40, 579 (1998)] is analyzed from a general viewpoint. We show that the perturbation gives rise to an effective potential for solitons, which consists of wells and barriers. We calculate the effect of these potential features on the solitons, establishing a direct relationship between the maxima, minima, and curvature of the potential with soliton deformations. When the typical wavelength of the perturbation is of the order of the soliton width, these deformations are seen to correspond to the excitation of shape modes and can lead to the dissipation of the soliton kinetic energy and, further, to the impossibility of soliton propagation. Thus, we demonstrate that the mechanism for length-scale competition is related to changes in the dynamics of the internal modes. We study different examples where the perturbation is introduced parametrically and nonparametrically to make it clear that our results apply to a wide class of equations.

We study the effects of the sequence on the propagation of nonlinear excitations in simple models of DNA, and how those effects are modified by noise. Starting from previous results on soliton dynamics on lattices defined by aperiodic potentials [23], we analyze the behavior of lattices built from real DNA sequences obtained from human genome data. We confirm the existence of threshold forces, already found in Fibonacci sequences, and of stop positions highly dependent on the specific sequence. Another relevant conclusion is that the effective potential, a collective coordinate formalism introduced by Salerno and Kivshar [21] is a useful tool to identify key regions that control the behaviour of a larger sequence. We then study how the fluctuations can assist the propagation process by helping the excitations to escape the stop positions. Our conclusions point out to improvements of the model which look promising to describe mechanical denaturation of DNA. Finally, we also consider how randomly distributed energy focus on the chain as a function of the sequence.

We study the effects of the genetic sequence on the propagation of nonlinear excitations in simple models of DNA in which we incorporate actual data from the human genome. We show that kink propagation requires forces over a certain threshold, a phenomenon already found for aperiodic sequences [F. Domínguez-Adame et al., Phys. Rev. E 52, 2183 (1995)]. For forces below threshold, the final stop positions are highly dependent on the specific sequence. Contrary to the conjecture advanced by Domínguez-Adame and co-workers, we find no evidence supporting the dependence of the kink dynamics on the information content of the genetic sequences considered. We discuss possible reasons for that result as well as its practical consequences. Physically, the results of our model are consistent with the stick-slip dynamics of the unzipping process observed in experiments. We also show that the effective potential, a collective coordinate formalism introduced by Salerno and Kivshar [Phys. Lett. A 193, 263 (1994)], is a useful tool to identify key regions in DNA that control the dynamical behavior of large segments. As a side result, we extend the previous studies on aperiodic sequences by analyzing the effect of the initial position of the kink, leading to further insight on the phenomenology observed in such systems.

Community ecology has traditionally relied on the competitive exclusion principle, a piece of common wisdom in conceptual frameworks developed to describe species assemblages. Key concepts in community ecology, such as limiting similarity and niche partitioning, are based on competitive exclusion. However, this classical paradigm in ecology relies on implications derived from simple, deterministic models. Here we show how the predictions of a symmetric, deterministic model about the way extinctions proceed can be utterly different from the results derived from the same model when ecological drift (demographic stochasticity) is explicitly considered. Using analytical approximations to the steady-state conditional probabilities for assemblages with two and three species, we demonstrate that stochastic competitive exclusion leads to a cascade of extinctions, whereas the symmetric, deterministic model predicts a multiple collapse of species. To test the robustness of our results, we have studied the effect of environmental stochasticity and relaxed the species symmetry assumption. Our conclusions highlight the crucial role of stochasticity when deriving reliable theoretical predictions for species community assembly.

In this work we analyze the possibility that the soliton dynamics in a simple nonlinear model allows functionally relevant predictions of the behaviour of DNA. This suggestion was first put forward by Salerno [M. Salerno, Phys. Rev. A 44 (1991) 5292] by showing results indicating that sine–Gordon kinks were set in motion at certain regions of a DNA sequence that include promoters. We revisit that system and show that the observed behaviour has nothing to do with promoters; on the contrary, it originates from the bases at the boundary, which are not part of the genome studied. We explain this phenomenology in terms of an effective potential for the kink center. This is further extended to disprove recent claims that the dynamics of kinks [E. Lennholm, M. H¨ornquist, Physica D 177 (2003) 233] or breathers [J.D. Bashford, J. Biol. Phys. 32 (2006) 27] has functional significance.We conclude that no such information can be extracted from this simple nonlinear model or its associated effective potentialThis work was supported by the Ministerio de Educación y Ciencia of Spain through grants BFM2003-07749- C05-01, FIS2004-01001, NAN2004-09087-C03-03, FIS2005- 973, by Comunidad de Madrid grant SIMUMAT-CM and by the Junta de Andaluc´ıa under projects 00481 and FQM-0207. S.C. is supported by a fellowship from the Consejería de Educación de la Comunidad de Madrid and the Fondo Social Europeo.Publicad