Search

Does every smooth vector field on a closed three-manifold, for example the three-sphere, have a closed orbit? The answer is no, according to counterexamples by K. Kuperberg and others. On the other hand, there is a special class of vector fields, called Reeb vector fields, which are associated to contact forms. The three-dimensional case of the Weinstein conjecture asserts that every Reeb vector field on a closed oriented three-manifold has a closed orbit. This conjecture was recently proved by Taubes using Seiberg-Witten theory. We give an introduction to the Weinstein conjecture, the main ideas in Taubes's proof, and the bigger picture into which it fits.

To purchase or authenticate to the full-text of this article, please visit this link: http://dx.doi.org/10.1111/j.1365-2745.2007.01210.x Byline: MARINA SEMCHENKO, MICHAEL J. HUTCHINGS (*), ELIZABETH A. JOHN (*) Keywords: Avena sativa; activated carbon; pot partitions; root exudates; root interactions; self-inhibition; self/non-self discrimination; substrate volume; tragedy of the commons Abstract: Summary It has been claimed that, compared with plants grown without competition, plants competing for a common pool of soil-based resources overproduce roots at the expense of reproduction (known as the tragedy of the commons). However, experiments on this phenomenon have manipulated not only the presence/absence of neighbours, but also substrate volume. Restricted substrate volume can itself affect plant growth, possibly through chemical self-inhibition of root growth. We conducted an experiment with oats (Avena sativa) to examine whether the experimental design used in previous studies on the tragedy of the commons in root competition might have confounded the effects of detection of neighbours and substrate volume. Six treatments combined two factors, namely the presence or absence of activated carbon, and either the presence of a plastic or a mesh partition, or the absence of a partition, between two plants in a pot. Activated carbon was used to adsorb root exudates and reduce their potential effects on root growth. In a seventh treatment, plants were grown alone in pots with half the substrate volume replaced by gravel, to fragment the distribution of available resources. We observed no tragedy of the commons in a comparison of the performance of plants grown with and without partitions; plants performed equally well in the presence and absence of root competition. In the treatment with gravel, plants displayed reduced tillering and shoot growth per unit root mass, and an earlier switch to reproduction. Pot partitioning was associated with inhibition of root growth that was mediated by root exudates. When activated carbon was present, plants in partitioned pots performed better than plants growing with a root competitor. We conclude that two processes could determine plant growth in the experimental design used in studies of the tragedy of the commons: (i) greater root self-inhibition in the more limited space of partitioned pots, and (ii) inefficient root placement in larger substrate volumes in unpartitioned pots that are shared with roots of a competitor. These findings provide a new challenge for experimental designs attempting to demonstrate the role of self/non-self discrimination in root competition. Author Affiliation: (*)School of Life Sciences, University of Sussex, Falmer, Brighton, Sussex, BN1 9QG, UK Article History: Received 25 June 2006 revision accepted 6 November 2006Handling Editor: David Gibson Article note: Correspondence: Marina Semchenko (e-mail marina.semchenko@ut.ee).