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Dominance relationships are a defining feature of the social organization of many animal species. Populations structured by absolute dominance usually maintain a generally linear hierarchy, while relative dominance occurs, for example, within territorial populations where an animal is likely to be dominant within its territory. Because relative dominance is dependent on social context, the traits associated with it are often unclear. Green anole lizards, Anolis carolinensis, are an ideal system in which to determine dominance-related traits, as anoles use territorial defence in most natural environments, but establish a dominance hierarchy at high densities such as those that occur in captivity. We hypothesized that anoles use similar morphological and behavioural traits to determine social status under both forms of social organization. To test this, we studied a natural population of anoles to determine the traits most predictive of male territory size and quality (as measured by the number of females overlapping a male's territory). While these measures of territory may be related, they measure different components of territorial success. We then used mathematical ranking algorithms to quantify dominance in a tournament of paired arena trials, and identified traits associated with rank. Our results showed that lizards with wider heads had higher social rank, while those with longer heads were more successful at territorial defence. We also found that, independently of morphology, lizards who behaved more aggressively ranked higher in dominance trials, although behaviour did not predict measures of territory. Together, our results indicate that different traits may determine absolute and relative dominance in the green anole.

Multiple models are discussed for ranking teams in a league and introduce a new model called the Oracle method. This is a Markovian method that can be customized to incorporate multiple team traits into its ranking. Using a foresight prediction of NFL game outcomes for the 2002-2013 seasons, it is shown that the Oracle method correctly picked 64.1% of the games under consideration, which is higher than any of the methods compared, including ESPN Power Rankings, Massey, Colley, and PageRank.

We consider continuous triangular maps on I N , where I is a compact interval in the Euclidean space R. We show, under some conditions, that the orbit of every point in a triangular map converges to a fixed point if and only if there are no periodic orbit of prime period two. As a consequence we obtain a result on global stability, namely, if there are no periodic orbits of prime period 2 and the triangular map has a unique fixed point, then the fixed point is globally asymptotically stable. We also discuss examples and applications of our results to competition models.