Computing the distribution of the maximum in balls-and-boxes problems, with application to clusters of disease cases

We present a rapid method for the exact calculation of the cumulative distribution function of the maximum of multinomially distributed random variables. The method runs in time $O(mn)$, where $m$ is the desired maximum and $n$ is the number of variables. We apply the method to the analysis of two situations where an apparent clustering of cases of a disease in some locality has raised the possibility that the disease might be communicable, and this possibility has been discussed in the recent literature. We conclude that one of these clusters may be explained on purely random grounds, whereas the other may not

Metastability and anomalous fixation in evolutionary games on scale-free networks

We study the influence of complex graphs on the metastability and fixation properties of a set of evolutionary processes. In the framework of evolutionary game theory, where the fitness and selection are frequency-dependent and vary with the population composition, we analyze the dynamics of snowdrift games (characterized by a metastable coexistence state) on scale-free networks. Using an effective diffusion theory in the weak selection limit, we demonstrate how the scale-free structure affects the system's metastable state and leads to anomalous fixation. In particular, we analytically and numerically show that the probability and mean time of fixation are characterized by stretched exponential behaviors with exponents depending on the network's degree distribution.Comment: 5 pages, 4 figures, to appear in Physical Review Letter

Stochasticity and evolutionary stability

In stochastic dynamical systems, different concepts of stability can be obtained in different limits. A particularly interesting example is evolutionary game theory, which is traditionally based on infinite populations, where strict Nash equilibria correspond to stable fixed points that are always evolutionarily stable. However, in finite populations stochastic effects can drive the system away from strict Nash equilibria, which gives rise to a new concept for evolutionary stability. The conventional and the new stability concepts may apparently contradict each other leading to conflicting predictions in large yet finite populations. We show that the two concepts can be derived from the frequency dependent Moran process in different limits. Our results help to determine the appropriate stability concept in large finite populations. The general validity of our findings is demonstrated showing that the same results are valid employing vastly different co-evolutionary processes

An odontological study of ovicaprine herding strategies in the North Atlantic islands. The potential of dental enamel defects for identifying secondary product utilisation in an archaeological context.

Abstract: Recent debate concerning the suitability of mortality profile analysis for identifying secondary product utilisation within archaeozoological assemblages has prompted the search for alternative methodologies. This research explores the potential of using weaning age to provide insight into herding strategies in ovicaprines, determined through the prevalence of developmental enamel defects. A histological methodology was developed, adapted to the specific nature of sheep molars through an understanding of formation processes and enamel structures. This established a relationship between weaning and developmental defects in modern sheep, revealed as distinct patterns in defect distribution within the enamel. Based on historical/archaeological data a weaning age model was developed for the North Atlantic region by which herding strategies could be recognised, specifically: mixed milk/meat subsistence, with an emphasis on milk (0-2 months) or on meat (2-4 months), and the optimisation of meat and/or wool (4-6 months). This methodology was then tested on archaeological material to interpret husbandry at Iron Age and Norse/Viking period sites. The results of this analysis showed that interpretations were in general agreement with those of mortality profile and correspondence analysis conducted as a methodological comparative. Some disparity, however, highlighted the ability of this new technique to provide more sensitivity in cases of mixed subsistence systems, possibly identifying the economic focus of husbandry, or where mortality profiles are confused. It was concluded that the study of weaning age has potential to provide valuable insight into ovicaprine husbandry in archaeological contexts, adding to the understanding of faunal assemblages, especially when supported with other evidence.Recent debate concerning the suitability of mortality profile analysis for identifying secondary product utilisation within archaeozoological assemblages has prompted the search for alternative methodologies. This research explores the potential of using weaning age to provide insight into herding strategies in ovicaprines, determined through the prevalence of developmental enamel defects. A histological methodology was developed, adapted to the specific nature of sheep molars through an understanding of formation processes and enamel structures. This established a relationship between weaning and developmental defects in modern sheep, revealed as distinct patterns in defect distribution within the enamel. Based on historical/archaeological data a weaning age model was developed for the North Atlantic region by which herding strategies could be recognised, specifically: mixed milk/meat subsistence, with an emphasis on milk (0-2 months) or on meat (2-4 months), and the optimisation of meat and/or wool (4-6 months). This methodology was then tested on archaeological material to interpret husbandry at Iron Age and Norse/Viking period sites. The results of this analysis showed that interpretations were in general agreement with those of mortality profile and correspondence analysis conducted as a methodological comparative. Some disparity, however, highlighted the ability of this new technique to provide more sensitivity in cases of mixed subsistence systems, possibly identifying the economic focus of husbandry, or where mortality profiles are confused. It was concluded that the study of weaning age has potential to provide valuable insight into ovicaprine husbandry in archaeological contexts, adding to the understanding of faunal assemblages, especially when supported with other evidence

An exactly solvable coarse-grained model for species diversity

We present novel analytical results about ecosystem species diversity that stem from a proposed coarse grained neutral model based on birth-death processes. The relevance of the problem lies in the urgency for understanding and synthesizing both theoretical results of ecological neutral theory and empirical evidence on species diversity preservation. Neutral model of biodiversity deals with ecosystems in the same trophic level where per-capita vital rates are assumed to be species-independent. Close-form analytical solutions for neutral theory are obtained within a coarse-grained model, where the only input is the species persistence time distribution. Our results pertain: the probability distribution function of the number of species in the ecosystem both in transient and stationary states; the n-points connected time correlation function; and the survival probability, definned as the distribution of time-spans to local extinction for a species randomly sampled from the community. Analytical predictions are also tested on empirical data from a estuarine fish ecosystem. We find that emerging properties of the ecosystem are very robust and do not depend on specific details of the model, with implications on biodiversity and conservation biology.Comment: 20 pages, 4 figures. To appear in Journal of Statistichal Mechanic

Nongaussian fluctuations arising from finite populations: Exact results for the evolutionary Moran process

The appropriate description of fluctuations within the framework of evolutionary game theory is a fundamental unsolved problem in the case of finite populations. The Moran process recently introduced into this context [Nowak et al., Nature (London) 428, 646 (2004)] defines a promising standard model of evolutionary game theory in finite populations for which analytical results are accessible. In this paper, we derive the stationary distribution of the Moran process population dynamics for arbitrary $2\times{}2$ games for the finite size case. We show that a nonvanishing background fitness can be transformed to the vanishing case by rescaling the payoff matrix. In contrast to the common approach to mimic finite-size fluctuations by Gaussian distributed noise, the finite size fluctuations can deviate significantly from a Gaussian distribution.Comment: 4 pages (2 figs). Published in Physical Review E (Rapid Communications

Stochastic differential equations for evolutionary dynamics with demographic noise and mutations

We present a general framework to describe the evolutionary dynamics of an arbitrary number of types in finite populations based on stochastic differential equations (SDE). For large, but finite populations this allows to include demographic noise without requiring explicit simulations. Instead, the population size only rescales the amplitude of the noise. Moreover, this framework admits the inclusion of mutations between different types, provided that mutation rates, $\mu$, are not too small compared to the inverse population size 1/N. This ensures that all types are almost always represented in the population and that the occasional extinction of one type does not result in an extended absence of that type. For $\mu N\ll1$ this limits the use of SDE's, but in this case there are well established alternative approximations based on time scale separation. We illustrate our approach by a Rock-Scissors-Paper game with mutations, where we demonstrate excellent agreement with simulation based results for sufficiently large populations. In the absence of mutations the excellent agreement extends to small population sizes.Comment: 8 pages, 2 figures, accepted for publication in Physical Review

Preservation of information in a prebiotic package model

The coexistence between different informational molecules has been the preferred mode to circumvent the limitation posed by imperfect replication on the amount of information stored by each of these molecules. Here we reexamine a classic package model in which distinct information carriers or templates are forced to coexist within vesicles, which in turn can proliferate freely through binary division. The combined dynamics of vesicles and templates is described by a multitype branching process which allows us to write equations for the average number of the different types of vesicles as well as for their extinction probabilities. The threshold phenomenon associated to the extinction of the vesicle population is studied quantitatively using finite-size scaling techniques. We conclude that the resultant coexistence is too frail in the presence of parasites and so confinement of templates in vesicles without an explicit mechanism of cooperation does not resolve the information crisis of prebiotic evolution.Comment: 9 pages, 8 figures, accepted version, to be published in PR