In analogy with Newton's law of gravity, the gravity law assumes that the number of individuals T ij that move between locations i and j per unit time is proportional to some power of the population of the source (m i ) and destination (n j ) locations, and decays with the distance r ij between them as ,where ! and ! are adjustable exponents and the deterrence function is chosen to fit the empirical data. Occasionally T ij is interpreted as the probability rate of individuals !of traveling from i to j, or an effective coupling between the two locations 24 . Despite its widespread use, the gravity law has notable limitations:i) We lack a rigorous derivation of (1). While entropy maximization 25 leads to (1) with ! = " = 1 , it fails to offer the functional form of f(r).ii) Lacking theoretical guidance, practitioners use a range of deterrence functions (power law or exponential) and up to nine parameters to fit the empirical data 5,7,8,11,14 .iii) As (1) requires previous traffic data to fit the parameters [ ], it is unable to predict mobility in regions where we lack systematic traffic data, areas of major interest in modeling of infectious diseases.iv)The gravity law has systematic predictive discrepancies. Indeed, in Fig. 1a we highlight two pairs of counties with similar origin and destination populations and comparable distance, so according to (1) the flux between them should be the same. Yet, the US census (see SI) documents an order of magnitude difference between the two fluxes: only 6 individuals commute between the two Alabama counties, while 44 in Utah.v) Equation (1) predicts that the number of commuters increases without limit as we increase the destination population n j , yet the number of commuters cannot exceed the source population m i , highlighting the gravity law's analytical inconsistency (see SI, Sect. 4).vi) Being deterministic, the gravity law cannot account for fluctuations in the number of travelers between two locations.Motivated by these known limitations, alternative approaches like the intervening opportunity model 26 or the random utility model 27 (SI, Sect. 7) have been proposed.While derived from first principles, these models continue to contain context specific tunable parameters, and their predictive power is at best comparable to the gravity law 28 .Here we introduce a modelling framework that relies on first principles and overcomes the problems (i) -(vi) of the gravity law. While commuting is a daily process, its source and destination is determined by job selection, a decision made over longer timescales. Using the natural partition of a country into counties (for which commuting data are collected), we assume that job selection consists of two steps ( Fig. 1 b, c):An individual seeks job offers from all counties, including his/her home county.The number of employment opportunities in each county is proportional to the resident population, n, assuming that there is one job opening for every n jobs individuals. We capture the benefits of a potential employment opportunity...

The theory of island biogeography 1 asserts that an island or a local community approaches an equilibrium species richness as a result of the interplay between the immigration of species from the much larger metacommunity source area and local extinction of species on the island (local community). Hubbell 2 generalized this neutral theory to explore the expected steady-state distribution of relative species abundance (RSA) in the local community under restricted immigration. Here we present a theoretical framework for the unified neutral theory of biodiversity 2 and an analytical solution for the distribution of the RSA both in the metacommunity (Fisher's log series) and in the local community, where there are fewer rare species. Rare species are more extinction-prone, and once they go locally extinct, they take longer to re-immigrate than do common species. Contrary to recent assertions 3 , we show that the analytical solution provides a better fit, with fewer free parameters, to the RSA distribution of tree species on Barro Colorado Island, Panama 4 , than the lognormal distribution 5,6 .The neutral theory in ecology 2,7 seeks to capture the influence of speciation, extinction, dispersal and ecological drift on the RSA under the assumption that all species are demographically alike on a per capita basis. This assumption, while only an approximation 8-10 , appears to provide a useful description of an ecological community on some spatial and temporal scales 2,7 . More significantly, it allows the development of a tractable null theory for testing hypotheses about community assembly rules. However, until now, there has been no analytical derivation of the expected equilibrium distribution of RSA in the local community, and fits to the theory have required simulations 2 with associated problems of convergence times, unspecified stopping rules, and precision 3 .The dynamics of the population of a given species is governed by generalized birth and death events (including speciation, immigration and emigration). Let b n,k and d n,k represent the probabilities of birth and death, respectively, in the kth species with n individuals with b 21;k ¼ d 0;k ¼ 0: Let p n,k (t) denote the probability that the kth species contains n individuals at time t. In the simplest scenario, the time evolution of p n,k (t) is regulated by the master equation 11-13 dp n;k ðtÞ dt ¼ p nþ1;k ðtÞd nþ1;k þ p n21;k ðtÞb n21;k 2 p n;k ðtÞðb n;k þ d n;k Þ ð 1Þ which leads to the steady-state or equilibrium solution, denoted by P:for n . 0 and where P 0,k can be deduced from the normalization condition P n P n;k ¼ 1: Note that there is no requirement of Box 1Derivation of the RSA of the local communityWe study the dynamics within a local community following the mathematical framework of McKane et al. 27 , who studied a mean-field stochastic model for species-rich communities. In our context, the dynamical rules 2 governing the stochastic processes in the community are:(1) With probability 1-m, pick two individuals at random from the local communi...

Many biological processes, from cellular metabolism to population dynamics, are characterized by allometric scaling (power-law) relationships between size and rate. An outstanding question is whether typical allometric scaling relationships--the power-law dependence of a biological rate on body mass--can be understood by considering the general features of branching networks serving a particular volume. Distributed networks in nature stem from the need for effective connectivity, and occur both in biological systems such as cardiovascular and respiratory networks and plant vascular and root systems, and in inanimate systems such as the drainage network of river basins. Here we derive a general relationship between size and flow rates in arbitrary networks with local connectivity. Our theory accounts in a general way for the quarter-power allometric scaling of living organisms, recently derived under specific assumptions for particular network geometries. It also predicts scaling relations applicable to all efficient transportation networks, which we verify from observational data on the river drainage basins. Allometric scaling is therefore shown to originate from the general features of networks irrespective of dynamical or geometric assumptions.