Predicting continental-scale patterns of bird species richness with spatially explicit models.

The causes of global variation in species richness have been debated for nearly two centuries with no clear resolution in sight. Competing hypotheses have typically been evaluated with correlative models that do not explicitly incorporate the mechanisms responsible for biotic diversity gradients. Here, we employ a fundamentally different approach that uses spatially explicit Monte Carlo models of the placement of cohesive geographical ranges in an environmentally heterogeneous landscape. These models predict species richness of endemic South American birds (2248 species) measured at a continental scale. We demonstrate that the principal single-factor and composite (species-energy, water-energy and temperature-kinetics) models proposed thus far fail to predict (r(2) < or =.05) the richness of species with small to moderately large geographical ranges (first three range-size quartiles). These species constitute the bulk of the avifauna and are primary targets for conservation. Climate-driven models performed reasonably well only for species with the largest geographical ranges (fourth quartile) when range cohesion was enforced. Our analyses suggest that present models inadequately explain the extraordinary diversity of avian species in the montane tropics, the most species-rich region on Earth. Our findings imply that correlative climatic models substantially underestimate the importance of historical factors and small-scale niche-driven assembly processes in shaping contemporary species-richness patterns.

Swap and fill algorithms in null model analysis: rethinking the knight's tour.

Community assembly rules are often inferred from patterns in presence-absence matrices. A challenging problem in the analysis of presence-absence matrices has been to devise a null model algorithm to produce random matrices with fixed row and column sums. Previous studies by Roberts and Stone [(1990) Oecologia 83:560-567] and Manly [(1995) Ecology 76:1109-1115] used a "Sequential Swap" algorithm in which submatrices are repeatedly swapped to produce null matrices. Sanderson et al. [(1998) Oecologia 116:275-283] introduced a "Knight's Tour" algorithm that fills an empty matrix one cell at a time. In an analysis of the presence-absence matrix for birds of the Vanuatu islands, Sanderson et al. obtained different results from Roberts and Stone and concluded that "results from previous studies are generally flawed". However, Sanderson et al. did not investigate the statistical properties of their algorithm. Using simple probability calculations, we demonstrate that their Knight's Tour is biased and does not sample all unique matrices with equal frequency. The bias in the Knight's Tour arises because the algorithm samples exhaustively at each step before retreating in sequence. We introduce an unbiased Random Knight's Tour that tests only a small number of cells and retreats by removing a filled cell from anywhere in the matrix. This algorithm appears to sample unique matrices with equal frequency. The Random Knight's Tour and Sequential Swap algorithms generate very similar results for the large Vanuatu matrix, and for other presence-absence matrices we tested. As a further test of the Sequential Swap, we constructed a set of 100 random matrices derived from the Vanuatu matrix, analyzed them with the Sequential Swap, and found no evidence that the algorithm is prone to Type I errors (rejecting the null hypothesis too frequently). These results support the original conclusions of Roberts and Stone and are consistent with Gotelli's [(2000) Ecology 81:2606-2621] Type I and Type II error tests for the Sequential Swap. In summary, Sanderson et al.'s Knight's Tourgenerates large variances and does not sample matrices equiprobably. In contrast, the Sequential Swap generates results that are very similar to those of an unbiased Random Knight's Tour, and is not overly prone to Type I or Type II errors. We suggest that the statistical properties of proposed null model algorithms be examined carefully, and that their performance judged by comparisons with artificial data sets of known structure. In this way, Type I and Type II error frequencies can be quantified, and different algorithms and indices can be compared meaningfully.