The stabilizing effect of a random environment

It is shown that the lottery competition model permits coexistence in a stochastic environment, but not in a constant environment. Conditions for coexistence and competitive exclusion are determined. Analysis of these conditions shows that the essential requirements for coexistence are overlapping generations and fluctuating birth rates which ensure that each species has periods when it is increasing. It is found that a species may persist provided only that it is favored sufficiently by the environment during favorable periods independently of the extent to which the other species is favored during its favorable periods.Coexistence is defined in terms of the stochastic boundedness criterion for species persistence. Using the lottery model as an example this criterion is justified and compared with other persistence criteria. Properties of the stationary distribution of population density are determined for an interesting limiting case of the lottery model and these are related to stochastic boundedness. An attempt is then made to relate stochastic boundedness for infinite population models to the behavior of finite population models.     

General population growth models with Allee effects in a random environment

Allee effects on population growth are quite common in nature, usually studied through deterministic models with a specific growth rate function.In order to seek the qualitative behaviour of populations induced by such effects, one should avoid model-specific behaviours. So, we use as a basis a general deterministic model, i.e. a model with a general growth rate function, to which we add the effect on the growth rate of the random fluctuations in environmental conditions. The resulting model is the general stochastic differential equation (SDE) model that we propose here.We consider two possible cases, weak Allee effects and strong Allee effects, which lead to different qualitative behaviours of the model.We will study the model properties for both cases in terms of existence and uniqueness of the solution, extinction and stationary behaviour of the population. The two cases will be compared with each other and with the general density-dependent SDE model without Allee effects.We then consider as an example the particular case of the classic logistic model and an Allee effect version of it.     

Prediction of multi-locus inbreeding coefficients and relation to linkage disequilibrium in random mating populations

An algorithm to predict the level of identity by descent simultaneously at multiple loci is presented, which can in principle be extended to any number of loci. The model assumes a random mating population, with random association of haplotypes. The relationship is shown between coefficients of multi-locus identity or non-identity by descent and moments of multi-locus linkage disequilibrium. Thus, these moments can be computed from the multilocus identity or, using algorithms derived previously to predict the disequilibria moments, vice-versa. The results can be applied to predict multi-locus identity in, for example, gene mapping.     

Diffusion approximations for one-locus multi-allele kin selection,mutation and random drift in group-structured populations: a unifying approach to selection models in population genetics

Diffusion approximations are ascertained from a two-time-scale argument in the case of a group-structured diploid population with scaled viability parameters depending on the individual genotype and the group type at a single multi-allelic locus under recurrent mutation, and applied to the case of random pairwise interactions within groups. The main step consists in proving global and uniform convergence of the distribution of the group types in an infinite population in the absence of selection and mutation, using a coalescent approach. An inclusive fitness formulation with coefficient of relatedness between a focal individual J affecting the reproductive success of an individual I, defined as the expected fraction of genes in I that are identical by descent to one or more genes in J in a neutral infinite population, given that J is allozygous or autozygous, yields the correct selection drift functions. These are analogous to the selection drift functions obtained with pure viability selection in a population with inbreeding. They give the changes of the allele frequencies in an infinite population without mutation that correspond to the replicator equation with fitness matrix expressed as a linear combination of a symmetric matrix for allozygous individuals and a rank-one matrix for autozygous individuals. In the case of no inbreeding, the mean inclusive fitness is a strict Lyapunov function with respect to this deterministic dynamics. Connections are made between dispersal with exact replacement (proportional dispersal), uniform dispersal, and local extinction and recolonization. The timing of dispersal (before or after selection, before or after mating) is shown to have an effect on group competition and the effective population size. In memory of Sam Karlin.     

The biennial life strategy in a random environment

A discrete-time population model with two age classes is studied which describes the growth of biennial plants in a randomly varying environment. A fraction of the oldest age class delays its flowering each year. The solution of the model involves products of random matrices. We calculate the exact mean and variance of the long-run geometric growth rate assuming a gamma distribution for the random number of offspring per flowering plant after one year. It is shown, both by analytical calculation and numerical examples, that it is profitable for the population to delay its flowering, in the sense that the average growth rate increases and the extinction probability decreases. The optimal values of the flowering fraction depend upon the environmental and model parameters.     

Modelling animal growth in random environments: An application using nonparametric estimation

We study a stochastic differential equation growth model to describe individual growth in random environments. In particular, in this paper, we discuss the estimation of the drift and the diffusion coefficients using nonparametric methods for the case of nonequidistant data for several trajectories. We illustrate the methodology by using bovine growth data. Our goal is to assess: (i) if the parametric models (with specific functional forms for the drift and the diffusion coefficients) previously used by us to describe the evolution of bovine weight were adequate choices; (ii) whether some alternative specific parameterized functional forms of these coefficients might be suggested for further parametric analysis of this data.     

From local interactions to population dynamics in site-based models of ecology

A central problem in ecology is relating the interactions of individuals-described in terms of competition, predation, interference, etc.-to the dynamics of the populations of these individuals-in terms of change in numbers of individuals over time. Here, we address this problem for a class of site-based ecological models, where local interactions between individuals take place at a finite number of discrete resource sites over non-overlapping generations and, between generations, individuals move randomly between sites over the entire system. Such site-based models have previously been applied to a wide range of ecological systems: from those involving contest or scramble competition for resources to host-parasite interactions and meta-populations. We show how the population dynamics of site-based models can be accurately approximated by and understood through deterministic and stochastic difference equations. Conversely, we use the inverse of this approximation to show what implicit assumptions are made about individual interactions by modelling of population dynamics in terms of difference equations. To this end, we prove a useful and general theorem: that any model in our class of site-based models has a corresponding stochastic difference equation population model, by which it can be approximated. This theorem allows us to calculate long-term population dynamics, evolutionary stable strategies and, by extending our theory to account for large deviations, extinction probabilities for a wide range of site-based systems. Our methodology is then illustrated to various examples of between species competition, predator-prey interactions and co-operation.     

Fixed and random effects selection in linear and logistic models

We address the problem of selecting which variables should be included in the fixed and random components of logistic mixed effects models for correlated data. A fully Bayesian variable selection is implemented using a stochastic search Gibbs sampler to estimate the exact model-averaged posterior distribution. This approach automatically identifies subsets of predictors having nonzero fixed effect coefficients or nonzero random effects variance, while allowing uncertainty in the model selection process. Default priors are proposed for the variance components and an efficient parameter expansion Gibbs sampler is developed for posterior computation. The approach is illustrated using simulated data and an epidemiologic example.     

Equilibrium selection in evolutionary games with random matching of players

We discuss stochastic dynamics of populations of individuals playing games. Our models possess two evolutionarily stable strategies: an efficient one, where a population is in a state with the maximal payoff (fitness) and a risk-dominant one, where players are averse to risks. We assume that individuals play with randomly chosen opponents (they do not play against average strategies as in the standard replicator dynamics). We show that the long-run behavior of a population depends on its size and the mutation level.     

Stability results for delayed-recruitment models in population dynamics

This paper relates the stability properties of a class of delay-difference equations to those of an associated scalar difference equation. Simple but powerful conditions for testing global stability are presented which are independent of the length of the time delay involved. For models which do not have globally stable equilibria, estimates of stability regions are obtained. Some well known baleen whale models are used to illustrate the results.     

The many-demes limit for selection and drift in a subdivided population

A diffusion approximation is obtained for the frequency of a selected allele in a population comprised of many subpopulations or demes. The form of the diffusion is equivalent to that for an unstructured population, except that it occurs on a longer time scale when migration among demes is restricted. This many-demes diffusion limit relies on the collection of demes always being in statistical equilibrium with respect to migration and drift for a given allele frequency in the total population. Selection is assumed to be weak, in inverse proportion to the number of demes, and the results hold for any deme sizes and migration rates greater than zero. The distribution of allele frequencies among demes is also described.     

The biennial life strategy in a random environment

In a previous paper (J. Math. Biol. 26, 199–215 (1988)) we calculated the mean and variance of the long-run geometric growth rate of a discrete-time population model with two age classes in a random environment. The formula which was used in that paper as the starting point for the computation of the variance represents only the contribution of the one-period variances. Here we supplement these results by a calculation of the exact variance. All qualitative conclusions reached before are unaffected.     

Bayesian covariance selection in generalized linear mixed models

The generalized linear mixed model (GLMM), which extends the generalized linear model (GLM) to incorporate random effects characterizing heterogeneity among subjects, is widely used in analyzing correlated and longitudinal data. Although there is often interest in identifying the subset of predictors that have random effects, random effects selection can be challenging, particularly when outcome distributions are nonnormal. This article proposes a fully Bayesian approach to the problem of simultaneous selection of fixed and random effects in GLMMs. Integrating out the random effects induces a covariance structure on the multivariate outcome data, and an important problem that we also consider is that of covariance selection. Our approach relies on variable selection-type mixture priors for the components in a special Cholesky decomposition of the random effects covariance. A stochastic search MCMC algorithm is developed, which relies on Gibbs sampling, with Taylor series expansions used to approximate intractable integrals. Simulated data examples are presented for different exponential family distributions, and the approach is applied to discrete survival data from a time-to-pregnancy study.     

Detecting local residue environment similarity for recognizing near‐native structure models

We developed a new representation of local amino acid environments in protein structures called the Side‐chain Depth Environment (SDE). An SDE defines a local structural environment of a residue considering the coordinates and the depth of amino acids that locate in the vicinity of the side‐chain centroid of the residue. SDEs are general enough that similar SDEs are found in protein structures with globally different folds. Using SDEs, we developed a procedure called PRESCO (Protein Residue Environment SCOre) for selecting native or near‐native models from a pool of computational models. The procedure searches similar residue environments observed in a query model against a set of representative native protein structures to quantify how native‐like SDEs in the model are. When benchmarked on commonly used computational model datasets, our PRESCO compared favorably with the other existing scoring functions in selecting native and near‐native models. Proteins 2014; 82:3255–3272. © 2014 Wiley Periodicals, Inc.     

Empirical Bayes estimation of random effects parameters in mixed effects logistic regression models

We extend an approach for estimating random effects parameters under a random intercept and slope logistic regression model to include standard errors, thereby including confidence intervals. The procedure entails numerical integration to yield posterior empirical Bayes (EB) estimates of random effects parameters and their corresponding posterior standard errors. We incorporate an adjustment of the standard error due to Kass and Steffey (KS; 1989, Journal of the American Statistical Association 84, 717-726) to account for the variability in estimating the variance component of the random effects distribution. In assessing health care providers with respect to adult pneumonia mortality, comparisons are made with the penalized quasi-likelihood (PQL) approximation approach of Breslow and Clayton (1993, Journal of the American Statistical Association 88, 9-25) and a Bayesian approach. To make comparisons with an EB method previously reported in the literature, we apply these approaches to crossover trials data previously analyzed with the estimating equations EB approach of Waclawiw and Liang (1994, Statistics in Medicine 13, 541-551). We also perform simulations to compare the proposed KS and PQL approaches. These two approaches lead to EB estimates of random effects parameters with similar asymptotic bias. However, for many clusters with small cluster size, the proposed KS approach does better than the PQL procedures in terms of coverage of nominal 95% confidence intervals for random effects estimates. For large cluster sizes and a few clusters, the PQL approach performs better than the KS adjustment. These simulation results agree somewhat with those of the data analyses.     

Stochastic models for phytoplankton dynamics in Mediterranean Sea

In this paper, we review some results obtained from three one-dimensional stochastic models, which were used to analyze picophytoplankton dynamics in two sites of the Mediterranean Sea. Firstly, we present a stochastic advection–reaction–diffusion model to describe the vertical spatial distribution of picoeukaryotes in a site of the Sicily Channel. The second model, which is an extended version of the first one, is used to obtain the vertical stationary profiles of two groups of picophytoplankton, i.e. Pelagophytes and Prochlorococcus, in the same marine site as in the previous case. Here, we include intraspecific competition of picophytoplanktonic groups for limiting factors, i.e. light intensity and nutrient concentration. Finally, we analyze the spatio-temporal behaviour of five picophytoplankton populations in a site of the Tyrrhenian Sea by using a reaction–diffusion–taxis model. The study is performed, taking into account the seasonal changes of environmental variables, obtained starting from experimental findings. The multiplicative noise source, present in all three models, mimics the random fluctuations of temperature and velocity field. The vertical profiles of chlorophyll concentration obtained from the stochastic models show a good agreement with experimental data sampled in the two marine sites considered. The results could be useful to devise a new class of models based on a stochastic approach and able to predict future changes in biomass primary production.     

Best linear prediction of breeding values in a forest tree improvement program

Summary Best Linear Prediction (BLP) was used to predict breeding values for 1,396 parents from progeny test data in an operational slash pine breeding program. BLP rankings of parents were compared to rankings of averaged standard scores, a common approach in forestry. Using BLP rankings, selection of higher ranking parents tends to choose parents in a larger number of more precise progeny tests. The trend is the opposite with standard scores; higher ranking parents tend to be those in fewer, less precise tests. BLP and a related methodology, Best Linear Unbiased Prediction (BLUP), were developed by dairy cattle breeders and have not been used widely outside of animal breeding for predicting breeding values from messy progeny test data. Application of either of these techniques usually requires simplifying assumptions to keep the problem computationally tractable. The more appropriate technique for a given application depends upon which set of assumptions are better for the given problem. An assumption of homogeneous genetic and error variances and covariances, generally made by animal breeders when applying BLUP, was inappropriate for our data. We employed an approach that treated fixed effects as known and treated the same trait measured in different environments as different traits with heterogeneous variance structures. As tree improvement programs become more complex, the ease with which BLP and BLUP handle messy data and incorporate diverse sources of information should make these techniques appealing to forest tree breeders.     

Population models in a periodically fluctuating environment

Summary We investigate the behavior of population models in the presence of a periodically fluctuating environment. We consider in particular single-species models and models of interspecific competition. It is shown that the fluctuations cause constant equilibrium states to be replaced by periodic equilibrium states, with a shift in the mean value relative to the constant-environment state. It is shown also that the locations of points of exchange of stability may be changed as a result of the fluctuations.     

Restricted best linear unbiased prediction and a selection model

Summary An equivalence between restricted best linear unbiased prediction (and thus restricted selection index) and a particular example of a selection model is presented. Specifically, the equivalence is between restricted selection and a model of selection on the residuals of the general mixed linear model. This result illustrates that restricted selection acts by nonrandomly sampling those genes that act pleiotropically in multiple trait genetic models. An expression for a mixed linear model which includes restrictions is also presented.