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Axelrod's model for culture dissemination offers a nontrivial answer to the question of why there is cultural diversity given that people's beliefs have a tendency to become more similar to each other's as they interact repeatedly. The answer depends on the two control parameters of the model, namely, the number F of cultural features that characterize each agent, and the number q of traits that each feature can take on, as well as on the size A of the territory or, equivalently, on the number of interacting agents. Here, we investigate the dependence of the number C of distinct coexisting cultures on the area A in Axelrod's model, the culture-area relationship, through extensive Monte Carlo simulations. We find a non-monotonous culture-area relation, for which the number of cultures decreases when the area grows beyond a certain size, provided that q is smaller than a threshold value qc = qc (F) and F > or = 3. In the limit of infinite area, this threshold value signals the onset of a discontinuous transition between a globalized regime marked by a uniform culture (C = 1), and a completely polarized regime where all C = qF possible cultures coexist. Otherwise, the culture- area relation exhibits the typical behavior of the species- area relation, i.e., a monotonically increasing curve the slope of which is steep at first and steadily levels off at some maximum diversity value.

In the last three decades, researchers have tried to identify universal patterns in the structure of food webs. It was recently proposed that the exponent eta characterizing the efficiency of the transport of energy in large and small food webs might have a universal value (eta = 1.13). In this work we establish lower and upper bounds for this exponent in a general spanning tree with a fixed number of trophic species and levels. When the number of species is large, the lower and upper bounds are equal to 1, implying that the result eta = 1.13 is due to finite-size effects and that the value of this exponent depends on the size of the web. We also evaluate analytically and numerically the exponent eta for hierarchical and random networks. In all cases the exponent eta depends on the number of trophic species K, and when K is large we have that eta-->1. Moreover, this result holds for any fixed number M of trophic levels.

Allometric scaling is one of the most pervasive laws in biology. Its origin, however, is still a matter of dispute. Recent studies have established that maximum metabolic rate scales with an exponent larger than that found for basal metabolism. This unpredicted result sets a challenge that can decide which of the concurrent hypotheses is the correct theory. Here we show that both scaling laws can be deduced from a single network model. Besides the 3/4-law for basal metabolism, the model predicts that maximum metabolic rate scales as M 6/7 , maximum heart rate as M −1/7 , and muscular capillary density as M −1/7 , in agreement with data.PACS numbers: 87.10.+e,87.23.-n Metabolic rate B and body mass M are connected by the scaling relation B = aM b , where a is a constant and b is the allometric exponent. Kleiber's law [1] (b = 3/4) characterizes the basal metabolism of almost all organisms [2,3,4], including mammals and birds. The predominance of quarter-power scaling in biology is experimentally evident [2,3,4] and is supported by recent theories based on the design of resource distribution networks [5,6]. Maximum metabolic rate, however, scales with an exponent somewhat larger than 3/4 (b ≈ 0.86) [4,7,8,9,10,11]. This result raises the question whether the scaling of maximal metabolic rate is governed by mechanisms different from those determining basal metabolism [4,8,9]. Here we reformulate the ideas of West, Brown and Enquist [5] to show that both scaling laws can be deduced from a single network model. Although it was recently demonstrated that the exponent 3/4 cannot be consistently derived from the assumptions of West et al. [5,12], we show that, using a different set of assumptions and the appropriate impedances, the model correctly predicts the 3/4-law for basal metabolism.The basal and maximum metabolic rates set the limits of the energetic range of endothermic animals. Considerable effort has recently been invested to understand the scaling of these variables [4,5,6,7,8,9,10,11,12,13,14,15,16]. In particular, maximal performance is very informative about animal design since during strenuous exercise the circulatory and respiratory systems are stressed to their uttermost. Recent studies have established that maximal metabolic rate (MMR) scales with an exponent larger than that found for basal metabolism, suggesting that the scaling of MMR is determined by mechanisms different from those controlling basal metabolic rate (BMR) [4,8,9]. We show that this difference in the allometric exponent is a consequence of dynamic adaptations in the resource supply network, which are needed to cope with the distinct demands for oxygen and nutrients. Simple dimensional analysis linking BMR to heat loss through the surface (claimed to support a 2/3-law for basal metabolism) [12] and simple geometric arguments [6] are unable to explain why different metabolic states have different allometric exponents.West, Brown and Enquist (WBE) proposed that the ubiquity of the 3/4-law lies in the resource distribution network common ...