Abstract1. Global concern about human impact on biological diversity has triggered an intense research agenda on drivers and consequences of biodiversity change in parallel with international policy seeking to conserve biodiversity and associated ecosystem functions. Quantifying the trends in biodiversity is far from trivial, however, as recently documented by meta-analyses, which report little if any net change in local species richness through time.2. Here, we summarise several limitations of species richness as a metric of biodiversity change and show that the expectation of directional species richness trends under changing conditions is invalid. Instead, we illustrate how a set of species turnover indices provide more information content regarding temporal trends in biodiversity, as they reflect how dominance and identity shift in communities over time.3. We apply these metrics to three monitoring datasets representing different ecosystem types. In all datasets, nearly complete species turnover occurred, but this was disconnected from any species richness trends. Instead, turnover was strongly influenced by changes in species presence (identities) and dominance (abundances).We further show that these metrics can detect phases of strong compositionalThis is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

The vertical distribution of phytoplankton is of fundamental importance for the dynamics and structure of aquatic communities. Here, using an advection-reaction-diffusion model, we investigate the distribution and competition of phytoplankton species in a water column, in which inverse resource gradients of light and a nutrient can limit growth of the biomass. This problem poses a challenge for ecologists, as the location of a production layer is not fixed, but rather depends on many internal parameters and environmental factors. In particular, we study the influence of an upper mixed layer (UML) in this system and show that it leads to a variety of dynamic effects: (i) Our model predicts alternative density profiles with a maximum of biomass either within or below the UML, thereby the system may be bistable or the relaxation from an unstable state may require a long-lasting transition. (ii) Reduced mixing in the deep layer can induce oscillations of the biomass; we show that a UML can sustain these oscillations even if the diffusivity is less than the critical mixing for a sinking phytoplankton population. (iii) A UML can strongly modify the outcome of competition between different phytoplankton species, yielding bistability both in the spatial distribution and in the species composition. (iv) A light limited species can obtain a competitive advantage if the diffusivity in the deep layers is reduced below a critical value. This yields a subtle competitive exclusion effect, where the oscillatory states in the deep layers are displaced by steady solutions in the UML. Finally, we present a novel graphical approach for deducing the competition outcome and for the analysis of the role of a UML in aquatic systems.

A dispersing billiard (Lorentz gas) and focusing billiards (in the form of a stadium) with time-dependent boundaries are considered. The problem of particle acceleration in such billiards is studied. For the Lorentz gas two cases of the time dependence are investigated: stochastic perturbations of the boundary and its periodic oscillations. Two types of focusing billiards with periodically forced boundaries are explored: a stadium with strong chaotic properties and a nearrectangle stadium. It is shown that in all cases billiard particles can reach unbounded velocities. Average velocities of the particle ensemble as functions of time and the number of collisions are obtained.

There is still considerable debate about which mechanisms drive the relationship between biodiversity and ecosystem function (BEF). Although most scientists agree on the existence of two underlying mechanisms, complementarity and selection, experimental studies keep producing contrasting results on the relative contributions of the two effects. We present a spatially explicit resource competition model and investigate how the strength of these effects is influenced by trait and environmental variability, resource distribution, and species pool size. Our results demonstrate that the increase of biomass production with increasing species numbers depends on the concurrence of environmental and trait variability: BEF relationships are stronger if functionally different species coexist in a landscape with heterogeneous resource supply. These large biodiversity effects arise from complementarity effects, whereas selection effects are maximized when broad trait ranges coincide with narrow ranges of resource supply ratios. Our results will therefore help to resolve the debate on complementarity and selection mechanisms.

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