The latitudinal gradient in species richness, wherein species richness peaks near the equator and declines toward the poles, is a widely recognized phenomenon that holds true for many taxa in all habitat types. Understanding the causative mechanism or mechanisms that generate the latitudinal gradient in species richness (LGSR) has been a major challenge, and the gradient remains unexplained. A different latitudinal trend (named ''Rapoport's rule''), in which the mean size of species geographical ranges tends to decline toward the equator, has been hypothesized by G. C. Stevens to play a key role in generating the LGSR when coupled with a version of the ''rescue effect,'' in which local populations toward the fringes of geographical ranges are sustained by immigration. The Stevens hypothesis is now commonly cited as a potential explanation for the LGSR and has provoked numerous empirical studies in macroecology and biogeography. However, important aspects of the hypothesis are not obvious in Stevens's verbal model and may go unrecognized, despite their major implications for empirical work related to large-scale ecological and evolutionary processes. Here we present mathematical simulation models that test the logical structure of the Stevens hypothesis, examine effects on global patterns of species richness produced by the mechanisms (Rapoport's rule and the rescue effect) explicitly identified by Stevens, and investigate the additional effect of competition.We find that Rapoport's rule on its own generates an LGSR opposite that of the real world, with species richness peaking at the poles rather than at the equator. The same qualitative result (a ''reverse'' LGSR) appears when rescue-effect regions, as described by Stevens, are added to the model. Building upon Stevens's verbal model, we then develop an explicit version of competition and show that competition alone tends to equalize species richness across all latitudes. However, when both Rapoport's rule and competition are included in the model, we find that a qualitatively correct LGSR is produced. Unlike previous hypotheses regarding the LGSR, this version of the model does not rely on a latitudinal gradient in the intensity of competition to produce an LGSR. However, detection of this LGSR depends on the spatial scale at which species richness is sampled, with the LGSR appearing only with regional, not local, sampling. In contrast, when competition is explicitly added to the model with both Rapoport's rule and the rescue effect, an LGSR that is qualitatively consistent with that of the real world does appear in both local and regional samples. This expanded version of the Stevens hypothesis potentially could explain the real-world LGSR, but all three elements (Rapoport's rule, the rescue effect, and competition) are crucial and must operate sufficiently strongly and in specific ways. The LGSR becomes apparent in the model only when parameter values for Rapoport's rule and the rescue effect are large, possibly unrealistically so, and when all points on Eart...

The latitudinal gradient in species richness, wherein species richness peaks near the equator and declines toward the poles, is a widely recognized phenomenon that holds true for many taxa in all habitat types. Understanding the causative mechanism or mechanisms that generate the latitudinal gradient in species richness (LGSR) has been a major challenge, and the gradient remains unexplained. A different latitudinal trend (named “Rapoport’s rule”), in which the mean size of species geographical ranges tends to decline toward the equator, has been hypothesized by G. C. Stevens to play a key role in generating the LGSR when coupled with a version of the “rescue effect,” in which local populations toward the fringes of geographical ranges are sustained by immigration. The Stevens hypothesis is now commonly cited as a potential explanation for the LGSR and has provoked numerous empirical studies in macroecology and biogeography. However, important aspects of the hypothesis are not obvious in Stevens’s verbal model and may go unrecognized, despite their major implications for empirical work related to large‐scale ecological and evolutionary processes. Here we present mathematical simulation models that test the logical structure of the Stevens hypothesis, examine effects on global patterns of species richness produced by the mechanisms (Rapoport’s rule and the rescue effect) explicitly identified by Stevens, and investigate the additional effect of competition. We find that Rapoport’s rule on its own generates an LGSR opposite that of the real world, with species richness peaking at the poles rather than at the equator. The same qualitative result (a “reverse” LGSR) appears when rescue‐effect regions, as described by Stevens, are added to the model. Building upon Stevens’s verbal model, we then develop an explicit version of competition and show that competition alone tends to equalize species richness across all latitudes. However, when both Rapoport’s rule and competition are included in the model, we find that a qualitatively correct LGSR is produced. Unlike previous hypotheses regarding the LGSR, this version of the model does not rely on a latitudinal gradient in the intensity of competition to produce an LGSR. However, detection of this LGSR depends on the spatial scale at which species richness is sampled, with the LGSR appearing only with regional, not local, sampling. In contrast, when competition is explicitly added to the model with both Rapoport’s rule and the rescue effect, an LGSR that is qualitatively consistent with that of the real world does appear in both local and regional samples. This expanded version of the Stevens hypothesis potentially could explain the real‐world LGSR, but all three elements (Rapoport’s rule, the rescue effect, and competition) are crucial and must operate sufficiently strongly and in specific ways. The LGSR becomes apparent in the model only when parameter values for Rapoport’s rule and the rescue effect are large, possibly unrealistically so, and when all points on Earth are ...

We use Bayesian methods to explore fitting the von Bertalanffy length model to tag-recapture data. We consider two popular parameterizations of the von Bertalanffy model. The first models the data relative to age at first capture; the second models in terms of length at first capture. Using data from a rainbow trout Oncorhynchus mykiss study we explore the relationship between the assumptions and resulting inference using posterior predictive checking, cross validation and a simulation study. We find that untestable hierarchical assumptions placed on the nuisance parameters in each model can influence the resulting inference about parameters of interest. Researchers should carefully consider these assumptions when modeling growth from tag-recapture data.

Thirty-two cases of glossopharyngeal neuralgia complicated by syncope, cardiac arrhythmias or convulsions, singly or together, have been reported in the world literature. A further case is described and the clinical features of these thirty-three are reviewed. It is recommended that treatment should be undertaken as a matter of urgency. In the first place, Carbamezapine, with often the addition of Atropine, may prove effective. However, surgical intervention appears to give a better chance of permanent relief. Four alternative methods of surgery are discussed and the cervical or the intracranial approach recommended. Surgery should not be delayed in patients who fail to respond to medical treatment or in whom recurrence of symptoms occurs.

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