Reduction of a metapopulation genetic model to an effective one island model

We explore a model of metapopulation genetics which is based on a more ecologically motivated approach than is frequently used in population genetics. The size of the population is regulated by competition between individuals, rather than by artificially imposing a fixed population size. The increased complexity of the model is managed by employing techniques often used in the physical sciences, namely exploiting time-scale separation to eliminate fast variables and then constructing an effective model from the slow modes. Remarkably, an initial model with 2$\mathcal{D}$ variables, where $\mathcal{D}$ is the number of islands in the metapopulation, can be reduced to a model with a single variable. We analyze this effective model and show that the predictions for the probability of fixation of the alleles and the mean time to fixation agree well with those found from numerical simulations of the original model.Comment: 16 pages, 4 figures. Supplementary material: 22 pages, 3 figure

The Clumping Transition in Niche Competition: a Robust Critical Phenomenon

We show analytically and numerically that the appearance of lumps and gaps in the distribution of n competing species along a niche axis is a robust phenomenon whenever the finiteness of the niche space is taken into account. In this case depending if the niche width of the species $\sigma$ is above or below a threshold $\sigma_c$, which for large n coincides with 2/n, there are two different regimes. For $\sigma > sigma_c$ the lumpy pattern emerges directly from the dominant eigenvector of the competition matrix because its corresponding eigenvalue becomes negative. For $\sigma </- sigma_c$ the lumpy pattern disappears. Furthermore, this clumping transition exhibits critical slowing down as $\sigma$ is approached from above. We also find that the number of lumps of species vs. $\sigma$ displays a stair-step structure. The positions of these steps are distributed according to a power-law. It is thus straightforward to predict the number of groups that can be packed along a niche axis and it coincides with field measurements for a wide range of the model parameters.Comment: 16 pages, 7 figures; http://iopscience.iop.org/1742-5468/2010/05/P0500

Number of Common Sites Visited by N Random Walkers

We compute analytically the mean number of common sites, W_N(t), visited by N independent random walkers each of length t and all starting at the origin at t=0 in d dimensions. We show that in the (N-d) plane, there are three distinct regimes for the asymptotic large t growth of W_N(t). These three regimes are separated by two critical lines d=2 and d=d_c(N)=2N/(N-1) in the (N-d) plane. For d<2, W_N(t)\sim t^{d/2} for large t (the N dependence is only in the prefactor). For 2<d<d_c(N), W_N(t)\sim t^{\nu} where the exponent \nu= N-d(N-1)/2 varies with N and d. For d>d_c(N), W_N(t) approaches a constant as t\to \infty. Exactly at the critical dimensions there are logaritmic corrections: for d=2, we get W_N(t)\sim t/[\ln t]^N, while for d=d_c(N), W_N(t)\sim \ln t for large t. Our analytical predictions are verified in numerical simulations.Comment: 5 pages, 3 .eps figures include

The shape of ecological networks

We study the statistics of ecosystems with a variable number of co-evolving species. The species interact in two ways: by prey-predator relationships and by direct competition with similar kinds. The interaction coefficients change slowly through successful adaptations and speciations. We treat them as quenched random variables. These interactions determine long-term topological features of the species network, which are found to agree with those of biological systems.Comment: 4 pages, 2 figure

Stochastic models in population biology and their deterministic analogs

In this paper we introduce a class of stochastic population models based on "patch dynamics". The size of the patch may be varied, and this allows one to quantify the departures of these stochastic models from various mean field theories, which are generally valid as the patch size becomes very large. These models may be used to formulate a broad range of biological processes in both spatial and non-spatial contexts. Here, we concentrate on two-species competition. We present both a mathematical analysis of the patch model, in which we derive the precise form of the competition mean field equations (and their first order corrections in the non-spatial case), and simulation results. These mean field equations differ, in some important ways, from those which are normally written down on phenomenological grounds. Our general conclusion is that mean field theory is more robust for spatial models than for a single isolated patch. This is due to the dilution of stochastic effects in a spatial setting resulting from repeated rescue events mediated by inter-patch diffusion. However, discrete effects due to modest patch sizes lead to striking deviations from mean field theory even in a spatial setting.Comment: 47 pages, 9 figure

The functional significance of dental and mandibular reduction in Homo: A catarrhine perspective.

The reduction in dental size and mandibular robusticity is regarded as a major trend in human evolution, traditionally considered the result of the peculiar extra-oral food processing skills of Homo. The use of stone tools and fire would have allowed our ancestors to chew softer food in smaller bite size, thus relaxing the selective pressures to keep a large dentition and a robust lower jaw. This perspective assumes that differences in dental size and mandibular robusticity in hominins represent functional dissimilarities. This study uses a catarrhine comparative approach to test this fundamental assumption of the hypotheses on dental and mandibular reduction in Homo. A sample of extant catarrhines and fossil hominins was used to test for correlations between dental size, mandibular robusticity, and dietary proxies, the latter include diet quality, diet heterogeneity, feeding time, and microwear variables. The effects of phylogeny and body size were considered. Findings support the association between technological developments in Homo and reduction in incisor size and mandibular corpus robusticity, though not for premolar, molar size, and symphyseal robusticity. These results challenge the functional interpretation of postcanine reduction and symphyseal changes in the genus Homo

Incorporating fine-scale environmental heterogeneity into broad-extent models

A key aim of ecology is to understand the drivers of ecological patterns, so that we can accurately predict the effects of global environmental change. However, in many cases, predictors are measured at a finer resolution than the ecological response. We therefore require data aggregation methods that avoid loss of information on fine-grain heterogeneity. We present a data aggregation method that, unlike current approaches, reduces the loss of information on fine-grain spatial structure in environmental heterogeneity for use with coarse-grain ecological datasets. Our method contains three steps: (a) define analysis scales (predictor grain, response grain, scale-of-effect); (b) use a moving window to calculate a measure of variability in environment (predictor grain) at the process-relevant scale (scale-of-effect); and (c) aggregate the moving window calculations to the coarsest resolution (response grain). We show the theoretical basis for our method using simulated landscapes and the practical utility with a case study. Our method is available as the grainchanger r package. The simulations show that information about spatial structure is captured that would have been lost using a direct aggregation approach, and that our method is particularly useful in landscapes with spatial autocorrelation in the environmental predictor variable (e.g. fragmented landscapes) and when the scale-of-effect is small relative to the response grain. We use our data aggregation method to find the appropriate scale-of-effect of land cover diversity on Eurasian jay Garrulus glandarius abundance in the UK. We then model the interactive effect of land cover heterogeneity and temperature on G. glandarius abundance. Our method enables us quantify this interaction despite the different scales at which these factors influence G. glandarius abundance. Our data aggregation method allows us to integrate variables that act at varying scales into one model with limited loss of information, which has wide applicability for spatial analyses beyond the specific ecological context considered here. Key ecological applications include being able to estimate the interactive effect of drivers that vary at different scales (such as climate and land cover), and to systematically examine the scale dependence of the effects of environmental heterogeneity in combination with the effects of climate change on biodiversity

Unstable decay and state selection II

The decay of unstable states when several metastable states are available for occupation is investigated using path-integral techniques. Specifically, a method is described which allows the probabilities with which the metastable states are occupied to be calculated by finding optimal paths, and fluctuations about them, in the weak noise limit. The method is illustrated on a system described by two coupled Langevin equations, which are found in the study of instabilities in fluid dynamics and superconductivity. The problem involves a subtle interplay between non-linearities and noise, and a naive approximation scheme which does not take this into account is shown to be unsatisfactory. The use of optimal paths is briefly reviewed and then applied to finding the conditional probability of ending up in one of the metastable states, having begun in the unstable state. There are several aspects of the calculation which distinguish it from most others involving optimal paths: (i) the paths do not begin and end on an attractor, and moreover, the final point is to a large extent arbitrary, (ii) the interplay between the fluctuations and the leading order contribution are at the heart of the method, and (iii) the final result involves quantities which are not exponentially small in the noise strength. This final result, which gives the probability of a particular state being selected in terms of the parameters of the dynamics, is remarkably simple and agrees well with the results of numerical simulations. The method should be applicable to similar problems in a number of other areas such as state selection in lasers, activationless chemical reactions and population dynamics in fluctuating environments.Comment: 28 pages, 6 figures. Accepted for publication in Phys. Rev.

Decline in the strength of genetic controls on aspen environmental responses from seasonal to century‐long phenomena

Understanding intra-specific variation in climate sensitivity could improve the prediction of tree responses to climate change. We attempted to identify the degree of genetic control of tree phenology and growth of trembling aspen (Populus tremuloides Mchx.) in a natural stand of this species in northwestern Quebec. We mapped and genotyped 556 aspen trees growing within the plot, using seven nuclear microsatellite loci for clone identification. We selected 13 clones (n of trees per clone >5, in total 350 trees) and evaluated the explanatory power of clone identity in (a) variability of spring leaf phenology and (b) short- and long-term growth responses. The clone's identity explained 43% of the variability in spring leaf phenology, between 18% and 20% of variability in response to monthly climate variables significantly affecting growth, between 8% and 26% of growth response to insect outbreaks, and 12% in the long-term growth rates. Strong clonal control of aspen phenology and moderate control of growth responses to monthly weather do not result in an equally large impact on long-term growth rates. The result suggests an important role of environmental extremes and within community interactions as factors averaging aspen growth performance at the stand level

Evaluating the discriminatory power of EQ-5D, HUI2 and HUI3 in a US general population survey using Shannon’s indices

OBJECTIVES: To compare quantitatively the discriminatory power of the EQ-5D, HUI2 and HUI3 in terms of absolute and relative informativity, using Shannon's indices. METHODS: EQ-5D and HUI2/3 data completed by a sample of the general adult US population (N = 3,691) were used. Five dimensions allowed head-to-head comparison of informativity: Mobility/Ambulation; Anxiety/Depression/Emotion; Pain/Discomfort (EQ-5D; HUI2; HUI3); Self-Care (EQ-5D; HUI2); and Cognition (HUI2; HUI3). Shannon's index and Shannon's Evenness index were used to assess absolute and relative informativity, both by dimension and by instrument as a whole. RESULTS: Absolute informativity was highest for HUI3, with the largest differences in Pain/Discomfort and Cognition. Relative informativity was highest for EQ-5D, with the largest differences in Mobility/Ambulation and Anxiety/Depression/Emotion. Absolute informativity by instrument was consistently highest for HUI3 and lowest for EQ-5D, and relative informativity was highest for EQ-5D and lowest for HUI3. DISCUSSION: Performance in terms of absolute and relative informativity of the common dimensions of the three instruments varies over dimensions. Several dimensions are suboptimal: Pain/Discomfort (EQ-5D) seems too crude with only 3 levels, and the level descriptions of Ambulation (HUI3) and Self-Care (HUI2) could be improved. In absence of a formal measure, Shannon's indices provide useful measures for assessing discriminatory power of utility instrument