Efficient Simulation of Multivariate Binomial and Poisson Distributions

Power investigations, for example, in statistical procedures for the assessment of agreement among multiple raters often require the simultaneous simulation of several dependent binomial or Poisson distributions to appropriately model the stochastical dependencies between the raters' results. Regarding the rather large dimensions of the random vectors to be generated and the even larger number of interactions to be introduced into the simulation scenarios to determine all necessary information on their distributions' dependence stucture, one needs efficient and fast algorithms for the simulation of multivariate Poisson and binomial distributions. Therefore two equivalent models for the multivariate Poisson distribution are combined to obtain an algorithm for the quick implementation of its multivariate dependence structure. Simulation of the multivariate Poisson distribution then becomes feasible by first generating and then convoluting independent univariate Poisson variates with appropriate expectations. The latter can be computed via linear recursion formulae. Similar means for simulation are also considered for the binomial setting. In this scenario it turns out, however, that exact computation of the probability function is even easier to perform; therefore corresponding linear recursion formulae for the point probabilities of multivariate binomial distributions are presented, which only require information about the index parameter and the (simultaneous) success probabilities, that is the multivariate dependence structure among the binomial marginals.     

Characterization of the diatom flora in the Lower Mountain Fork (Oklahoma,USA), a novel regulated river with a disjunct population of the diatom Didymosphenia geminata (Bacillariophyta)

Regulated rivers are novel ecosystems with altered temperature and flow regimes that can be used to test distribution patterns of microscopic organisms, such as diatoms. Our objective was to describe the spatial and seasonal patterns of diatoms in a cold-water, oligotrophic river within a region of warm-water, mesotrophic rivers. The Lower Mountain Fork, in south-east Oklahoma (USA), is maintained as a year-around, stocked fishery by the release of cold, hypolimnetic water from Broken Bow Lake and is the southern-most known site of Didymosphenia geminata in North America. Epilithic diatoms were sampled six times at nine sites over a distance of 15.5 km and, within this area, 27 times at the site of the main Didymosphenia bloom. Percentage composition data were analysed for assemblage composition using multivariate analysis, nutrient specificity using a diatom-based metric, and species associations using similarity profiles. Eighty-eight taxa were found, of which 10 were unidentifiable and included local undescribed species and species clusters. Three species [Gomphonema (parvulum morph), Achnanthidium rivulare and Achnanthidium minutissimum] comprised over 60% of the diatom abundance at all sites, and downstream and seasonal patterns were evident for both these and less abundant taxa. Notably, diatom assemblages in the three sites below the dam were similar to that at the lowermost site, below a much smaller dam. The oligotraphentic diatom assemblage reflected the water chemistry of the river. The Didymosphenia bloom had been scoured by a large spate prior to the study and the species was present at two of the nine sites in low numbers but failed to bloom during the study, possibly because of a trend towards increasing phosphorus concentrations in the reservoir (Didymosphenia blooms under low P concentrations). No other species shared Didymosphenia’s distribution pattern over the study reach, highlighting the novelty of Didymosphenia’s presence in the river.     

Estimating plant abundance using inflated beta distributions: Applied learnings from a lichen–caribou ecosystem

Quantifying abundance and distribution of plant species can be difficult because data are often inflated with zero values due to rarity or absence from many ecosystems. Terrestrial fruticose lichens (Cladonia and Cetraria spp.) occupy a narrow ecological niche and have been linked to the diets of declining caribou and reindeer populations (Rangifer tarandus) across their global distribution, and conditions related to their abundance and distribution are not well understood. We attempted to measure effects related to the occupancy and abundance of terrestrial fruticose lichens by sampling and simultaneously modeling two discrete conditions: absence and abundance. We sampled the proportion cover of terrestrial lichens at 438 vegetation plots, including 98 plots having zero lichens. A zero‐inflated beta regression model was employed to simultaneously estimate both the absence and the proportion cover of terrestrial fruticose lichens using fine resolution satellite imagery and light detection and ranging (LiDAR) derived covariates. The probability of lichen absence significantly increased with shallower groundwater, taller vegetation, and increased Sphagnum moss cover. Vegetation productivity, Sphagnum moss cover, and seasonal changes in photosynthetic capacity were negatively related to the abundances of terrestrial lichens. Inflated beta regression reliably estimated the abundance of terrestrial lichens (R² = .74) which was interpolated on a map at fine resolution across a caribou range to support ecological conservation and reclamation. Results demonstrate that sampling for and simultaneously estimating both occupancy and abundance offer a powerful approach to improve statistical estimation and expand ecological inference in an applied setting. Learnings are broadly applicable to studying species that are rare, occupy narrow niches, or where the response variable is a proportion value containing zero or one, which is typical of vegetation cover data.     

Effects of climate change on the distribution of Iberian tree species

Question: Will the predicted climate changes affect species distribution in the Iberian Peninsula? Location: Iberian Peninsula (Spain and Portugal). Methods: We modelled current and future tree distributions as a function of climate, using a computational framework that made use of one machine learning technique, the random forest (RF) algorithm. This algorithm provided good predictions of the current distribution of each species, as shown by the area under the corresponding receiver operating characteristics (ROC) curves. Species turnover, richness and the change in distributions over time to 2080 under four Intergovernmental panel on climate change (IPCC) scenarios were calculated using the species map outputs. Results and Conclusions: The results show a notable reduction in the potential distribution of the studied species under all the IPCC scenarios, particularly so for mountain conifer species such as Pinus sylvestris, P. uncinata and Abies alba. Temperate species, especially Fagus sylvatica and Quercus petraea, were also predicted to suffer a reduction in their range; also sub‐mediterranean species, especially Q. pyrenaica, were predicted to undergo notable decline. In contrast, typically Mediterranean species appeared to be generally more capable of migration, and are therefore likely to be less affected.     

Evolution and ecology of lizard body sizes

Aim Body size is instrumental in influencing animal physiology, morphology, ecology and evolution, as well as extinction risk. I examine several hypotheses regarding the influence of body size on lizard evolution and extinction risk, assessing whether body size influences, or is influenced by, species richness, herbivory, island dwelling and extinction risk. Location World‐wide. Methods I used literature data and measurements of museum and live specimens to estimate lizard body size distributions. Results I obtained body size data for 99% of the world's lizard species. The body size–frequency distribution is highly modal and right skewed and similar distributions characterize most lizard families and lizard assemblages across biogeographical realms. There is a strong negative correlation between mean body size within families and species richness. Herbivorous lizards are larger than omnivorous and carnivorous ones, and aquatic lizards are larger than non‐aquatic species. Diurnal activity is associated with small body size. Insular lizards tend towards both extremes of the size spectrum. Extinction risk increases with body size of species for which risk has been assessed. Main conclusions Small size seems to promote fast diversification of disparate body plans. The absence of mammalian predators allows insular lizards to attain larger body sizes by means of release from predation and allows them to evolve into the top predator niche. Island living also promotes a high frequency of herbivory, which is also associated with large size. Aquatic and nocturnal lizards probably evolve large size because of thermal constraints. The association between large size and high extinction risk, however, probably reflects a bias in the species in which risk has been studied.     

Subpopulations of Cryptocephalus beetles (Coleoptera: Chrysomelidae): geographically close but genetically far

Abstract. The leaf beetles Cryptocephalus coryli, C. decemmaculatus and C. nitidulus are of conservation concern and are included on the UK Biodiversity Action Plan. The distinctiveness of the disjunct remaining populations of these beetles was compared to that of more continuously distributed Cryptocephalus species. This was carried out with a view to defining evolutionary significant units (ESUs) in the rare species. A portion of the cytochrome b gene, an intergenic spacer and partial tRNA was analysed from 93 specimens of Cryptocephalus beetle (Coleoptera: Chrysomelidae). Considerable sequence divergence was apparent in all the species, even at an intersite scale when the distances between sampled localities were very small (< 1 km). Intrapopulation, intersite and interpopulation divergence observed in the rare species was reflected in the species that have a more continuous distribution, implying that dispersal ability in these species is poor and gene flow can be impeded by relatively trivial barriers to dispersal. The evidence suggests that the disjunct populations of the rare Cryptocephalus species can, tentatively, be considered as ESUs. This has important implications for management strategies and reintroductions.     

On the impact of correlation between collaterally consanguineous cells on lymphocyte population dynamics

During an adaptive immune response, lymphocytes proliferate for five to twenty-five cell divisions, then stop and die over a period of weeks. Based on extensive flow cytometry data, Hawkins et al. (Proc Natl Acad Sci USA 104:5032–5037, 2007) introduced a cell-level stochastic model of lymphocyte population dynamics, called the Cyton Model, that accurately captures mean lymphocyte population size as a function of time. In Subramanian et al. (J Math Biol 56(6):861–892, 2008), we performed a branching process analysis of the Cyton Model and deduced from parameterizations for in vitro and in vivo data that the immune response is predictable despite each cell’s fate being highly variable. One drawback of flow cytometry data is that individual cells cannot be tracked, so that it is not possible to investigate dependencies in the fate of cells within family trees. In the absence of this information, while the Cyton Model abandons one of the usual assumptions of branching processes (the independence of lifetime and progeny number), it adopts another of the standard branching processes hypotheses: that the fates of progeny are stochastically independent. However, new experimental observations of lymphocytes show that the fates of cells in the same family tree are not stochastically independent. Hawkins et al. (2008, submitted) report on ciné lapse photography experiments where every founding cell’s family tree is recorded for a system of proliferating lymphocytes responding to a mitogenic stimulus. Data from these experiments demonstrate that the death-or-division fates of collaterally consanguineous cells (those in the same generation within a founding cell’s family tree) are strongly correlated, while there is little correlation between cells of distinct generations and between cells in distinct family trees. As this finding contrasts with one of the assumptions of the Cyton Model, in this paper we introduce three variants of the Cyton Model with increasing levels of collaterally consanguineous correlation structure to incorporate these new found dependencies. We investigate their impact on the predicted expected variability of cell population size. Mathematically we conclude that while the introduction of correlation structure leaves the mean population size unchanged from the Cyton Model, the variance of the population size distribution is typically larger. Biologically, through comparison of model predictions for Cyton Model parameterizations determined by in vitro and in vivo experiments, we deduce that if collaterally consanguineous correlation extends beyond cousins, then the immune response is less predictable than would be concluded from the original Cyton Model. That is, some of the variability seen in data that we previously attributed to experimental error could be due to intrinsic variability in the cell population size dynamics.      

Energetics and dynamics of global integrals modeling interaction between stiff filaments

The attractive and spacing interaction between pairs of filaments via cross-linkers, e.g. myosin oligomers connecting actin filaments, is modeled by global integral kernels for negative binding energies between two intersecting stiff and long rods in a (projected) two-dimensional situation, for simplicity. Whereas maxima of the global energy functional represent intersection angles of ‘minimal contact’ between the filaments, minima are approached for energy values tending to −∞, representing the two degenerate states of parallel and anti-parallel filament alignment. Standard differential equations of negative gradient flow for such energy functionals show convergence of solutions to one of these degenerate equilibria in finite time, thus called ‘super-stable’ states. By considering energy variations under virtual rotation or translation of one filament with respect to the other, integral kernels for the resulting local forces parallel and orthogonal to the filament are obtained. For the special modeling situation that these variations only activate ‘spring forces’ in direction of the cross-links, explicit formulas for total torque and translational forces are given and calculated for typical examples. Again, the two degenerate alignment states are locally ‘super-stable’ equilibria of the assumed over-damped dynamics, but also other stable states of orthogonal arrangement and different asymptotic behavior can occur. These phenomena become apparent if stochastic perturbations of the local force kernels are implemented as additive Gaussian noise induced by the cross-link binding process with appropriate scaling. Then global filament dynamics is described by a certain type of degenerate stochastic differential equations yielding asymptotic stationary processes around the alignment states, which have generalized, namely bimodal Gaussian distributions. Moreover, stochastic simulations reveal characteristic sliding behavior as it is observed for myosin-mediated interaction between actin filaments. Finally, the forgoing explicit and asymptotic analysis as well as numerical simulations are extended to the more realistic modeling situation with filaments of finite length.

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