1. We measured the body sizes (weights or lengths) of animal species found in the food webs of natural communities. In c. 90% of the feeding links among the animal species with known sizes, a larger predator consumes a smaller prey. 2. Larger predators eat prey with a wider range of body sizes than do smaller predators. The geometric mean predator size increases with the size of prey. The increase in geometric mean predator size is less than proportional to the increase in prey size (i.e. has a slope less than 1 on log-log coordinates). 3. The geometric mean sizes of prey and predators increase as the habitat of webs changes from aquatic to terrestrial to coastal to marine. Within each type of habitat, mean prey sizes are always less than mean predator sizes, and prey and predator sizes are always positively correlated. 4. Feeding relations order the metabolic types of organisms from invertebrate to vertebrate ectotherm to vertebrate endotherm. Organisms commonly eat other organisms with the same or lower metabolic type, but (with very rare exceptions) organisms do not eat other organisms with a higher metabolic type. Mean sizes of prey increase as the metabolic type of prey changes from invertebrate to vertebrate ectotherm to vertebrate endotherm, but the same does not hold true for predators. 5. Prey and predator sizes are positively correlated in links from invertebrate prey to invertebrate predators. In links with other combinations of prey and predator metabolic types, the correlation between prey and predator body sizes is rarely large when it is positive, and in some cases is even negative. 6. Species sizes are roughly log-normally distributed. 7. Body size offers a good (though not perfect) interpretation of the ordering of animal species assumed in the cascade model, a stochastic model of food web structure. When body size is taken as the physical interpretation of the ordering assumed in the cascade model, and when the body sizes of different animal species are taken as log-normally distributed, many of the empirical findings can be explained in terms of the cascade model.

In this article we consider a size structured population model with a nonlinear growth rate depending on the individual's size and on the total population. Our purpose is to take into account the competition for a resource (as it can be light or nutrients in a forest) in the growth of the individuals and study the influence of this nonlinear growth in the population dynamics. We study the existence and uniqueness of solutions for the model equations, and also prove the existence of a (compact) global attractor for the trajectories of the dynamical system defined by the solutions of the model. Finally, we obtain sufficient conditions for the convergence to a stationary size distribution when the total population tends to a constant value, and consider some simple examples that allow us to know something about their global dynamics.

We present the derivation of the continuous-time equations governing the limit dynamics of discrete-time reaction-diffusion processes defined on heterogeneous metapopulations. We show that, when a rigorous time limit is performed, the lack of an epidemic threshold in the spread of infections is not limited to metapopulations with a scale-free architecture, as it has been predicted from dynamical equations in which reaction and diffusion occur sequentially in time. DOI: 10.1103/PhysRevE.78.012902 PACS number͑s͒: 87.23.Ϫn, 89.75.Hc, 89.75.Fb The analysis of the spread of infectious diseases on complex networks has become a central issue in modern epidemiology ͓1͔ and, indeed, it was one of the main motivations for the development of percolation theory ͓2͔. While the initial approach was focussed on local contact networks ͓3-6͔, i.e., social networks within single populations ͑cities, urban areas͒, a new approach has been recently introduced for dealing with the spread of diseases in ensembles of ͑local͒ populations with a complex spatial arrangement and connected by migration ͓7͔. Such sets of connected populations living in a patchy environment are called metapopulations in ecology, and their study began in 1967 with the theory of island biogeography ͓8͔.In some recent models of epidemic spreading, the location of the patches in space is treated explicitly thanks to the increasing of computational power ͑see, for instance, ͓9͔͒. In ͓7,10,11͔, however, an alternative approach based on the formalism used in the statistical mechanics of complex networks is presented. Precisely, the topology of the spatial network of local populations ͑nodes͒ is mathematically encoded by means of the connectivity ͑degree͒ distribution p͑k͒, defined as the probability that a randomly chosen node has degree k. Moreover, each node contains two types of particles: A particles corresponding to susceptible individuals, and B particles that correspond to infected ones. Within each node, a transmission process ͑reaction͒ occurs between particles of different type, and migratory flows take place among nodes ͑diffusion͒ at constant rates. Therefore, although the detailed description of the spatial network is lost, the approach offers an elegant description of the epidemic spread in terms of densities of A particles and B particles in patches of degree k at time t, here denoted by A,k ͑t͒ and B,k ͑t͒ respectively. The reaction and diffusion ͑RD͒ processes modeling the spread of an infectious disease are considered as a two-step process in ͓7,11͔. First, inside each network node, the reaction takes place under the assumption of a homogenous mixing and conserving the total number of particles. In particular, the spread of the infection within a population follows the dynamics of a susceptible-infected-susceptible model which is described by the reactionscorresponding to the recovering ͑at a rate ͒ and transmission ͑at a rate ␤͒ processes, respectively. Second, after the reaction, fixed fractions 0 ഛ D A ഛ 1 and 0 ഛ D B ഛ 1 of each type of par...

This paper is devoted to the analysis of the early dynamics of an SIS epidemic model defined on networks. The model, introduced by Gross, D'Lima and Blasius in 2006, is based on the pair-approximation formalism and assumes that, at a given rewiring rate, susceptible nodes replace an infected neighbour by a new susceptible neighbour randomly selected among the pool of susceptible nodes in the population. The analysis uses a pair closure that improves the widely assumed in epidemic models defined on regular and homogeneous networks, and applies it to better understand the early epidemic spread on Poisson, exponential, and (truncated) scale-free networks. Two extinction scenarios, one dominated by transmission and the other one by rewiring, are characterized by considering the limit system of the model equations close to the beginning of the epidemic. Moreover, an analytical condition on the model parameters for the occurrence of a bistability region is obtained.

We present a study of the continuous-time equations governing the dynamics of a susceptible-infectedsusceptible model on heterogeneous metapopulations. These equations have been recently proposed as an alternative formulation for the spread of infectious diseases in metapopulations in a continuous-time framework. Individual-based Monte Carlo simulations of epidemic spread in uncorrelated networks are also performed revealing a good agreement with analytical predictions under the assumption of simultaneous transmission or recovery and migration processes.