Body size and latitudinal gradients in regional diversity of New World birds

Axe latitudinal gradients in regional diversity random or biased with respect to body size? Using data for the New World avifauna, I show that the slope of the increase in regional species richness from the Arctic to the equator is not independent of body size. The increase is steepest among small and medium-sized species, and shallowest among the largest species. This is reflected in latitudinal variation in the shape of frequency distributions of body sizes in regional subsets of the New World avifauna. Because species are added disproportionately in small and medium size classes towards low latitudes, distributions become less widely spread along the body size axis than expected from the number of species. These patterns suggest an interaction between the effects of latitude and body size on species richness, implying that mechanisms which vary with both latitude and body size may be important determinants of high tropical diversity in New World birds.

Argyrodes: phylogeny, sociality and interspecific interactions - a report on the Argyrodes symposium, Badplaas 2001

Argyrodes Simon 1864 is a large, cosmopolitan theridiid genus whose members exhibit a wide range of foraging techniques which usually involve exploiting other spiders, either by using their webs, stealing their food, or preying on them directly. We held a symposium on this genus at the 15th International Congress of Arachnology, Badplaas, South Africa in order to obtain a clearer perspective on the relationship between the phylogeny of the genus and the different foraging techniques. We concluded that Argyrodes forms a monophyletic group within the Theridiidae, and that there are clear monophyletic clades within the genus (already identified as species groups) that appear to share behavioral characteristics. We found no clear indication that foraging behaviors such as kleptoparasitism (stealing food) evolved from araneophagy (eating spiders) or vice versa. However, it appears that species that specialize in either kleptoparasitism or araneophagy use additional techniques in comparison to species that readily use both foraging modes. During our examination of Argyrodes/host interactions we noted the importance of Nephila species as hosts of Argyrodes species around the world and the impact of Argyrodes on Nephila. We also noted the fluid nature of the relationship between Argyrodes and the spiders with which they interact. For example, an Argyrodes/host relationship can change to an Argyrodes/prey relationship, and the type of kleptoparasitic behavior employed by an Argyrodes can change when it changes host species. The importance of eating silk was also noted and identified as an area for further research. We concluded that more work involving international collaboration is needed to fully understand the phylogeny of the genus and the relationships between the different types of foraging behaviors.

Boundary value problems for systems of difference equations associated with systems of second-order ordinary differential equations

We study difference equations which arise as discrete approximations to two-point boundary value problems for systems of second-order ordinary differential equations. We formulate conditions which guarantee a priori bounds on first differences of solutions to the discretized problem. We establish existence results for solutions to the discretized boundary value problems subject to nonlinear boundary conditions. We apply our results to show that solutions to the discrete problem converge to solutions of the continuous problem in an aggregate sense. (C) 2002 Elsevier Science Ltd. All rights reserved.

Existence of multiple solutions for second-order discrete boundary value problems

We give conditions on f involving pairs of discrete lower and discrete upper solutions which lead to the existence of at least three solutions of the discrete two-point boundary value problem yk+1 - 2yk + yk-1 + f (k, yk, vk) = 0, for k = 1,..., n - 1, y0 = 0 = yn,, where f is continuous and vk = yk - yk-1, for k = 1,..., n. In the special case f (k, t, p) = f (t) greater than or equal to 0, we give growth conditions on f and apply our general result to show the existence of three positive solutions. We give an example showing this latter result is sharp. Our results extend those of Avery and Peterson and are in the spirit of our results for the continuous analogue. (C) 2002 Elsevier Science Ltd. All rights reserved.

Difference equations associated with fully nonlinear boundary value problems for second order ordinary differential equations

We study the continuous problem y"=f(x,y,y'), xc[0,1], 0=G((y(0),y(1)),(y'(0), y'(1))), and its discrete approximation (y(k+1)-2y(k)+y(k-1))/h(2) =f(t(k), y(k), v(k)), k = 1,..., n-1, 0 = G((y(0), y(n)), (v(1), v(n))), where f and G = (g(0), g(1)) are continuous and fully nonlinear, h = 1/n, v(k) = (y(k) - y(k-1))/h, for k =1,..., n, and t(k) = kh, for k = 0,...,n. We assume there exist strict lower and strict upper solutions and impose additional conditions on f and G which are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. We show that the discrete approximation also has solutions which approximate solutions of the continuous problem and converge to the solution of the continuous problem when it is unique, as the grid size goes to 0. Homotopy methods can be used to compute the solution of the discrete approximation. Our results were motivated by those of Gaines.