Probabilistic analysis of a genealogical model of animal group patterns

Many social animals live in stable groups, and it has been argued that kinship plays a major role in their group formation process. In this study we present the mathematical analysis of a recent model which uses kinship as a main factor to explain observed group patterns in a finite sample of individuals. We describe the average number of groups and the probability distribution of group sizes predicted by this model. Our method is based on the study of recursive equations underlying these quantities. We obtain asymptotic equivalents for probability distributions and moments as the sample size increases, and we exhibit power-law behaviours. Computer simulations are also utilized to measure the extent to which the asymptotic approximation can be applied with confidence.     

Understanding animal group-size distributions

One of the most striking aspects of animal groups is their remarkable variation in size, both within and between species. While a number of mechanistic models have been proposed to explain this variation, there are few comprehensive datasets against which these models have been tested. In particular, we only vaguely understand how environmental factors and behavioral activities affect group-size distributions. Here we use observations of House sparrows (Passer domesticus) to investigate the factors determining group-size distribution. Over a wide range of conditions, we observed that animal group sizes followed a single parameter distribution known as the logarithmic distribution. This single parameter is the mean group size experienced by a randomly chosen individual (including the individual itself). For sparrows, the experienced mean group size, and hence the distribution, was affected by four factors: morning temperature, place, behavior and the degree of food spillage. Our results further indicate that the sparrows regulate the mean group size they experience, either by groups splitting more or merging less when local densities are high. We suggest that the mean experienced group size provides a simple but general tool for assessing the ecology and evolution of grouping.     

Monte Carlo simulations of protein amyloid formation reveal origin of sigmoidal aggregation kinetics

Severe conditions and lack of cure for many amyloid diseases make it highly desired to understand the underlying principles of formation of fibrillar aggregates (amyloid). Here, amyloid formation from peptides was studied using Monte Carlo simulations. Systems of 20, 50, 100, 200 or 500 hexapeptides were simulated. Association kinetics were modeled equal for fibrillar and other (inter- and intra-peptide) contacts and assumed to be faster the lower the effective contact order, which represents the distance in space. Attempts to form contacts were thus accepted with higher probability the lower the effective contact order, whereby formation of new contacts next to preexisting ones is favored by shorter physical separation. Kinetic discrimination was invoked by using two different life-times for formed contacts. Contacts within amyloid fibrils were assumed to have on average longer life-time than other contacts. We find that the model produces fibrillation kinetics with a distinct lag phase, and that the fibrillar contacts need to dissociate on average 5-20 times slower than all other contacts for the fibrillar structure to dominate at equilibrium. Analysis of the species distribution along the aggregation process shows that no other intermediate is ever more populated than the dimer. Instead of a single nucleation event there is a concomitant increase in average aggregate size over the whole system, and the occurrence of multiple parallel processes makes the process more reproducible the larger the simulated system. The sigmoidal shape of the aggregation curves arises from cooperativity among multiple interactions within each pair of peptides in a fibril. A governing factor is the increasing probability as the aggregation process proceeds of neighboring reinforcing contacts. The results explain the very strong bias towards cross β-sheet fibrils in which the possibilities for cooperativity among interactions involving neighboring residues and the repetitive use of optimal side-chain interactions are explored at maximum.     

Integrating nested spatial scales: implications for the coexistence of competitors on a patchy resource

1. Intraspecific aggregation at a single spatial scale can promote the coexistence of competitors. This paper demonstrates how this same mechanism can be applied to the many systems that are patchy at two scales, with patches nested within 'superpatches'.
2. Data are presented from a field study showing that insects living in rotting fruits have aggregated distributions in the fruits under a single tree, and that the mean density and degree of aggregation varies significantly among trees. Observations in this system motivate the following models.
3. A model of competition has been developed between two species which explicitly represents spatial variation at two scales. By integrating the probability distributions for each scale, the marginal distributions of competitors over all patches can be found and used to calculate coexistence criteria. This model assumes global movement of the competitors.
4. Although spatial variation at a single scale may not be sufficient for coexistence, the total variation over all patches can allow coexistence. Variation in mean densities among superpatches and variation in the degree of aggregation among superpatches both promote coexistence, but act in different ways.
5. A second model of competition between two species is described which incorporates the effects of limited movement among superpatches. Limited movement among superpatches generally promotes coexistence, and also leads to correlations among aggregation and the mean densities of competitors.     

Estimating Intermittent Individual Spawning Behavior via Disaggregating Group Data

In order to understand fish biology and reproduction, it is important to know the fecundity patterns of individual fish, as frequently established by recording the output of mixed-sex groups of fish in a laboratory setting. However, for understanding individual reproductive health and modeling purposes it is important to estimate individual fecundity from group fecundity. We created a multistage method that disaggregates group-level data into estimates for individual-level clutch size and spawning interval distributions. The first stage of the method develops estimates of the daily spawning probability of fish. Daily spawning probabilities are then used to calculate the log likelihood of candidate distributions of clutch size. Selecting the best candidate distribution for clutch size allows for a Monte Carlo resampling of annotations of the original data which state how many fish spawned on which day. We verify this disaggregation technique by combining data from fathead minnow pairs, and checking that the disaggregation method reproduced the original clutch sizes and spawning intervals. This method will allow scientists to estimate individual clutch size and spawning interval distributions from group spawning data without specialized or elaborate experimental designs.     

Species-range size distributions in Britain

The detailed forms of species-range size distributions in Britain are determined and contrasted for ten taxonomic assemblages (liverworts, vascular plants, molluscs [aquatic and terrestrial], dragonflies, macro-moths. butterflies, birds [breeding and wintering], mammals). All are strongly right-skewed when range sizes are untransformed. A logarithmic transformation fails to normalise the distribution for all but one group, and the distributions for several groups are not readily normalised at all. Taxa with larger median range sizes have species-range size distributions that are less strongly right-skewed. The median observed range sizes of species in each of the taxonomic groups fall, in terms of decreasing range size, in the sequence wintering birds < breeding birds < mammals < butterflies < terrestrial molluscs < dragonflies < aquatic molluscs < vascular plants < moths < liverworts. Despite the difficulties in deriving a simple and sensible mechanistic model for range size distributions, this is likely to be the most important next step towards understanding their forms.     

Power-law versus exponential distributions of animal group sizes

There has been some confusion concerning the animal group size: an exponential distribution was deduced by maximizing the entropy; lognormal distributions were practically used; as power-law decay with exponent 3/2 was proposed in physical analogy to aerosol condensation. Here I show that the animal group-size distribution follows a power-law decay with exponent 1, and is truncated at a cut-off size which is the expected size of the groups an arbitrary individual engages in. An elementary model of animal aggregation based on binary splitting and coalescing on contingent encounter is presented. The model predicted size distribution holds for various data from pelagic fishes and mammalian herbivores in the wild.     

Statistical tests for taxonomic distinctiveness from observations of monophyly

The observation of monophyly for a specified set of genealogical lineages is often used to place the lineages into a distinctive taxonomic entity. However, it is sometimes possible that monophyly of the lineages can occur by chance as an outcome of the random branching of lineages within a single taxon. Thus, especially for small samples, an observation of monophyly for a set of lineages--even if strongly supported statistically--does not necessarily indicate that the lineages are from a distinctive group. Here I develop a test of the null hypothesis that monophyly is a chance outcome of random branching. I also compute the sample size required so that the probability of chance occurrence of monophyly of a specified set of lineages lies below a prescribed tolerance. Under the null model of random branching, the probability that monophyly of the lineages in an index group occurs by chance is substantial if the sample is highly asymmetric, that is, if only a few of the sampled lineages are from the index group, or if only a few lineages are external to the group. If sample sizes are similar inside and outside the group of interest, however, chance occurrence of monophyly can be rejected at stringent significance levels (P < 10(-5)) even for quite small samples (approximately 20 total lineages). For a fixed total sample size, rejection of the null hypothesis of random branching in a single taxon occurs at the most stringent level if samples of nearly equal size inside and outside the index group--with a slightly greater size within the index group--are used. Similar results apply, with smaller sample sizes needed, when reciprocal monophyly of two groups, rather than monophyly of a single group, is of interest. The results suggest minimal sample sizes required for inferences to be made about taxonomic distinctiveness from observations of monophyly.     

The size and demographic composition of social groups of wild orang-utans.

Mackinnon (1974) observed social sub-groups of wild orang-utans. The groups' distribution of size and age-sex composition can test mathematical models of group formation. As predicted by models published previously, a truncated Poisson distribution describes the group sizes well. The inferred parameters governing group formation conform to a pattern in other primate species. A simple model for the demographic composition of a sub-group assumes that an individual joins a sub-group with a probability independent of his age-sex class and of other members of the sub-group. This model's predictions, which agree poorly with observation, offer a quantitative baseline against which one can infer when a demographic combination occurs with a non-random frequency.     

The distribution of species range size: a stochastic process

The major role played by environmental factors in determining the geographical range sizes of species raises the possibility of describing their long-term dynamics in relatively simple terms, a goal which has hitherto proved elusive. Here we develop a stochastic differential equation to describe the dynamics of the range size of an individual species based on the relationship between abundance and range size, derive a limiting stationary probability model to quantify the stochastic nature of the range size for that species at steady state, and then generalize this model to the species-range size distribution for an assemblage. The model fits well to several empirical datasets of the geographical range sizes of species in taxonomic assemblages, and provides the simplest explanation of species-range size distributions to date.     

The distribution of hauled-out ringed seals and an interpretation of Taylor's law

Summary The sizes of the groups in which ringed seals (Phoca hispida) haul out were studied from aerial survey data from a wide area of the Canadian Arctic. Densities of hauled-out ringed seals varied over a wide range, but the mean group size was comparatively constant. Ringed seal distribution data obeyed Taylor's law of variation; about half the increase in aggregation with density was due to change in group size, and half to increasingly clumpy distribution of groups.     

Context-dependent group size choice in fish

The costs and benefits of group membership vary with the size of groups, and individuals are expected to modify their choice of groups in response to ecological factors such as food availability and predation risk. We experimentally examined context-dependent group size choice in a shoaling fish, the banded killifish, Fundulus diaphanus, by using nondirectional odour cues to simulate a food source or a successful attack by a predator (food or alarm treatments) in the laboratory. Group sizes were significantly smaller in the food treatment and larger in the alarm treatment than in control trials. When presented with food and alarm cues together, fish formed groups that were larger than control groups but smaller than those seen with alarm cues alone. These results are consistent with theoretical predictions based on the known benefits and costs of grouping and with previous laboratory work examining the individual shoal choice behaviour of single fish. To examine possible mechanisms of group formation, we developed an individual-based model of shoaling behaviour in which simulated fish were allowed to modify the area over which they interacted with neighbouring individuals. Group size distributions produced by the model were a good approximation of our experimental data. We suggest that local behavioural interaction rules of this type are a potential mechanism by which fish may individually adjust grouping behaviour without requiring extensive information on the position and movement of all possible shoalmates.     

Self-assemblage and quorum in the earthworm Eisenia fetida (Oligochaete, Lumbricidae)

Despite their ubiquity and ecological significance in temperate ecosystems, the behavioural ecology of earthworms is not well described. This study examines the mechanisms that govern aggregation behaviour specially the tendency of individuals to leave or join groups in the compost earthworm Eisenia fetida, a species with considerable economic importance, especially in waste management applications. Through behavioural assays combined with mathematical modelling, we provide the first evidence of self-assembled social structures in earthworms and describe key mechanisms involved in cluster formation. We found that the probability of an individual joining a group increased with group size, while the probability of leaving decreased. Moreover, attraction to groups located at a distance was observed, suggesting a role for volatile cues in cluster formation. The size of earthworm clusters appears to be a key factor determining the stability of the group. These findings enhance our understanding of intra-specific interactions in earthworms and have potential implications for extraction and collection of earthworms in vermicomposting processes.     

A Bayesian heterogeneous analysis of variance approach to inferring recent selective sweeps

The distribution of microsatellite allele sizes in populations aids in understanding the genetic diversity of species and the evolutionary history of recent selective sweeps. We propose a heterogeneous Bayesian analysis of variance model for inferring loci involved in recent selective sweeps by analyzing the distribution of allele sizes at multiple loci in multiple populations. Our model is shown to be consistent with a multilocus test statistic, ln RV, proposed for identifying microsatellite loci involved in recent selective sweeps. Our methodology differs in that it accepts original allele size data rather than summary statistics and allows the incorporation of prior knowledge about allele frequencies using a hierarchical prior distribution consisting of log normal and gamma probability distributions. Interesting features of the model are its ability to simultaneously analyze allele size data for any number of populations and to cope with the presence of any number of selected loci. The utility of the method is illustrated by application to two sets of microsatellite allele size data for a group of West African Anopheles gambiae populations. The results are consistent with the suppressed-recombination model of speciation, and additional candidate loci on chromosomes 2 (079 and 175) and 3 (088) are discovered that escaped former analysis.     

Effects of group size on aggregation against desiccation in woodlice (Isopoda: Oniscidea)

Aggregation in terrestrial isopods, a behaviour that results in the formation of dense clusters, is readily accepted as a mechanism of resistance to desiccation. Thus, aggregation is considered to be an adaptation to terrestrial life in this fully terrestrial suborder of crustaceans. In the present study of Porcellio scaber Latreille, a cosmopolitan species, individual water loss is investigated experimentally as a function of the size of the aggregates and, for the first time, over a large range of group sizes (groups of 1, 10, 20, 40, 60, 80 and 100 individuals). From the perspective of an isolated individual, aggregation behaviour is effective in reducing the rate of water loss whatever the group size, and reduces the individual water loss rate by more than half in large groups. However, the water loss rate of an individual follows a power law according to group size. Accordingly, if the addition of individuals to small groups strongly reduces the water losses per individual, adding individuals to large groups only slightly reduces the individual water losses. Thus, the successful reduction of the water loss rate by this aggregation behaviour is confirmed, although only up to a certain limit, particularly if the number of individuals per aggregate exceeds 50–60 under the experimental conditions used in the present study. Moreover, the individual surface area exposed to the air, as a function of group size, follows a similar pattern (i.e. a similar power law). Thus, a geometrical explanation is proposed for the nonlinear water losses in woodlice aggregates. These results are discussed in relation to the group sizes observed both in the laboratory and the field.     

Nondetection sampling bias in marked presence‐only data

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Complete Numerical Solution of the Diffusion Equation of Random Genetic Drift

A numerical method is presented to solve the diffusion equation for the random genetic drift that occurs at a single unlinked locus with two alleles. The method was designed to conserve probability, and the resulting numerical solution represents a probability distribution whose total probability is unity. We describe solutions of the diffusion equation whose total probability is unity as complete. Thus the numerical method introduced in this work produces complete solutions, and such solutions have the property that whenever fixation and loss can occur, they are automatically included within the solution. This feature demonstrates that the diffusion approximation can describe not only internal allele frequencies, but also the boundary frequencies zero and one. The numerical approach presented here constitutes a single inclusive framework from which to perform calculations for random genetic drift. It has a straightforward implementation, allowing it to be applied to a wide variety of problems, including those with time-dependent parameters, such as changing population sizes. As tests and illustrations of the numerical method, it is used to determine: (i) the probability density and time-dependent probability of fixation for a neutral locus in a population of constant size; (ii) the probability of fixation in the presence of selection; and (iii) the probability of fixation in the presence of selection and demographic change, the latter in the form of a changing population size.     

Evolution, Ecology, and Multimodal Distributions of Body Size

The sizes of organisms are determined by their interactions with their environment and related ecological and evolutionary processes. Recent studies of body size distributions across communities show evidence for multimodality. The multiple modes were originally explained as a consequence of textural discontinuities in habitat structure. Because communities consist of species that are drawn from lineages, body size patterns within lineages will affect those that are expressed in communities. We used a cellular automation model to argue that multimodality in body sizes within lineages can arise from a few fundamental evolutionary mechanisms alone. We tested the hypothesis using body size data for 138 fish genera and found strong support for the idea that evolution structures body size distributions. The results suggest, first, that we should expect the distribution of body sizes within lineages to be multimodal and second, that a coherent theory of community body size distributions will need to combine both evolutionary and ecological perspectives. Received 28 January 2002; accepted 21 March 2002