Isotopic variation complicates analysis of trophic relations within the fish community of Plußsee: a small, deep, stratifying lake

Analysis of carbon and nitrogen stable isotopes has allowed freshwater ecologists to examine lake food webs in increasing detail. Many such studies have highlighted the existence of separate within-lake pelagic and benthic-littoral food webs but are typically conducted on large (> 10 km2) lakes, whereas the majority of lakes are actually relatively small. We used stable isotope analysis (δ13C & δ15N) to examine trophic interactions between fish and their prey in Plu�see, as an example of a small, stratifying lake, and to determine whether separate pelagic/benthic-littoral food webs could be distinguished in such systems. Our results indicate that the Plu�see food web was complicated, and due to extensive intra-annual isotopic variation in zooplankton (e.g. cladoceran δ13C annual range = 25.6�), it may be impossible to definitively assign consumers from small, eutrophic stratified lakes to pelagic or benthic-littoral food webs. We present evidence that some components of the Plu�see food web (large bream) may be subsidised by carbon of methanogenic origin.

Graphs of relations and Hilbert series

We are discussing certain combinatorial and counting problems related to quadratic algebras. First we give examples which confirm the Anick conjecture on the minimal Hilbert series for algebras given by $n$ generators and $\frac {n(n-1)}{2}$ relations for $n \leq 7$. Then we investigate combinatorial structure of colored graph associated to relations of RIT algebra. Precise descriptions of graphs (maps) corresponding to algebras with maximal Hilbert series are given in certain cases. As a consequence it turns out, for example, that RIT algebra may have a maximal Hilbert series only if components of the graph associated to each color are pairwise 2-isomorphic.