Temporal, diel and spatial variability of decapod larvae from St Paul`s Rocks, an equatorial oceanic island of Brazil

Temporal, spatial and diel variation in the distribution and abundance of organisms is an inherent property of ecological systems. The present study describes these variations and the composition of decapod larvae from the surface waters of St Paul`s Rocks. The expeditions to the archipelago were carried out in April, August and November 2003, March 2004 and May 2005. Surface plankton samples were collected during the morning and dusk periods, inside the inlet and in increasing distances around the archipelago (similar to 150, 700 and 1500 m). The identification resulted in 51 taxa. Seven species, six genera and larvae of the families Pandalidae and Portunidae were identified for the first time in the area. The mean larval density varied from zero to 150.2 +/- 69.6 individuals 100 m(-3) in the waters surrounding the archipelago and from 1.7 +/- 3.0 to 12,827 +/- 15,073 individuals 100 m(-3) inside the inlet. Significant differences on larval density were verified between months and period of the day, but not among the three sites around the archipelago. Cluster and non-metric multidimensional scaling analysis indicated that the decapod larvae community was divided into benthic and pelagic assemblages. Indicator species analysis (ISA) showed that six Brachyura taxa were good indicators for the inlet, while three sergestids were the main species from the waters around the archipelago. These results suggest that St Paul`s Rocks can be divided into two habitats, based on larval composition, density and diversity values: the inlet and the waters surrounding the archipelago.

Bourgin-Yang version of the Borsuk-Ulam theorem for Z(pk)-equivariant maps

Let G = Z(pk) be a cyclic group of prime power order and let V and W be orthogonal representations of G with V-G = W-G = W-G = {0}. Let S(V) be the sphere of V and suppose f: S(V) -> W is a G-equivariant mapping. We give an estimate for the dimension of the set f(-1){0} in terms of V and W. This extends the Bourgin-Yang version of the Borsuk-Ulam theorem to this class of groups. Using this estimate, we also estimate the size of the G-coincidences set of a continuous map from S(V) into a real vector space W'.

Wecken type problems for self-maps of the Klein bottle

We consider various problems regarding roots and coincidence points for maps into the Klein bottle . The root problem where the target is and the domain is a compact surface with non-positive Euler characteristic is studied. Results similar to those when the target is the torus are obtained. The Wecken property for coincidences from to is established, and we also obtain the following 1-parameter result. Families which are coincidence free but any homotopy between and , , creates a coincidence with . This is done for any pair of maps such that the Nielsen coincidence number is zero. Finally, we exhibit one such family where is the constant map and if we allow for homotopies of , then we can find a coincidence free pair of homotopies.