Leis de escala alométricas para as taxas metabólicas interespecíficas e para cadeias alimentares - Lauro Antonio Barbosa (2007)
parameters and its effects are studied by a branch of biology called allometry. In this thesis we have studied, from the view points of physics and biology, two topics: (i) allometry of metabolic rates and (ii) scaling relations in food webs. (i) The allometric scaling relation B = aMb connects the metabolic rate B with the body mass M. Here a is a constant and b is the so-called allometric exponent. For many years it was a universal believe that the basal metabolic rate of all organisms is described by Kleibers law (b=3/4). In attempting to derive a fundamental explanation for Kleiber law, West, Brown and Enquist proposed an elegant model based on resource distribution networks common to all organisms. However, some mathematical mistakes have been found in this model, showing that the 3=4-law cannot be correctly derived. Recently, the 3/4 - law is cast in doubt even by experimental data. Moreover, it is known that maximum metabolic rate scales with an exponent larger than 3=4. This result raises the question whether the scaling of maximal metabolic rate is governed by mechanisms different from those determining basal metabolism. We reformulate the ideas ofWest, Brown and Enquist and show that this model furnishes the exponent of the maximum metabolic rate and that the exponent of the basal one can be obtained through an approximation. The difference between these exponents is due to dynamic adaptations in the resource supply network. Moreover, we present a new model where the metabolic states of organisms are described in a four-dimensional space (three biological lengths and a physiological time). We considered that in this space the mass and energy densities are size-invariant quantities (independent of body mass). The different metabolic states (basal and maximum) are described by assuming that the biological lengths and the characteristic time are connected by different dominant transport processes of energy and mass. The results for the maximum metabolism of animals agree with the empirical data. For the basal metabolism we find that the exponent is in the interval [2=3; 3=4] and probably has the value b = 0; 714. (ii) Food webs are diagrams showing the predation relationships among species in a community. In the last three decades, researchers have tried to identify universal patterns in the structure of food webs. Recently, Garlaschelli, Caldarelli and Pietronero have studied food web as transportation networks by extending to them the concept of allometric scaling, how branching properties change with network size. They have proposed that the exponent ´ characterizing the efficiency of the transport of material and energy in large and small food webs might have a universal value (´ = 1; 13). We establish a lower and upper bounds for this exponent in a general spanning tree with 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 ´ = 1; 13 is due to finite size effects and that the value of this exponent depends on the size of the web.