Variable effort fishing models in random environments

We study the growth of populations in a random environment subjected to variable effort fishing policies. The models used are stochastic differential equations and the environmental fluctuations may either affect an intrinsic growth parameter or be of the additive noise type. Density-dependent natural growth and fishing policies are of very general form so that our results will be model independent. We obtain conditions on the fishing policies for non-extinction and for non-fixation at the carrying capacity that are very similar to the conditions obtained for the corresponding deterministic model. We also obtain conditions for the existence of stationary distributions (as well as expressions for such distributions) very similar to conditions for the existence of an equilibrium in the corresponding deterministic model. The results obtained provide minimal requirements for the choice of a wise density-dependent fishing policy.     

Linear mixed models with flexible distributions of random effects for longitudinal data

Normality of random effects is a routine assumption for the linear mixed model, but it may be unrealistic, obscuring important features of among-individual variation. We relax this assumption by approximating the random effects density by the seminonparameteric (SNP) representation of Gallant and Nychka (1987, Econometrics 55, 363-390), which includes normality as a special case and provides flexibility in capturing a broad range of nonnormal behavior, controlled by a user-chosen tuning parameter. An advantage is that the marginal likelihood may be expressed in closed form, so inference may be carried out using standard optimization techniques. We demonstrate that standard information criteria may be used to choose the tuning parameter and detect departures from normality, and we illustrate the approach via simulation and using longitudinal data from the Framingham study.     

Polymorphism in random environments

A two-allele diploid model is described in which the fitnesses of the three genotypes are stationary stochastic processes. It is shown that a stable polymorphism will occur if the geometric mean fitness of the heterozygote exceeds that of both homozygotes. It is possible for the mean fitness of the population to be lower in polymorphic than in the associated monomorphic populations.     

Sex in random environments

Based on the theory of natural selection it is not obvious why sexual reproduction should evolve in Mendelian populations. Sexually reproducing organisms incur a “cost of meiosis”: an asexual lineage would grow at twice the rate of a comparable sexual lineage. A plausible and popular explanation for the widespread occurrence of sexual reproduction is that it adapts a lineage to temporal uncertainty in the environment. Computer simulation of a model introduced and partially analyzed in a companion paper (Hines & Moore,1981) suggests that under some of the hypothetical conditions, sexuality is advantageous, but the conditions are very restricted if only one or a few loci are selected. In the companion paper, to make analytical progress, it was necessary to assume small environmental effects or that the fitnesses of the homozygotes at each locus were identical in each generation, although fluctuating between generations. No such assumptions were made here. In addition the effect of an absorbing barrier was studied in the simulations.The computer model envisages from 1–4 loci, each with two alleles, selected independently. In each generation, each locus experiences one of three selection regimes chosen at random; each genotype is favored by one of the three selection regimes. The fitness of a multi-locus genotype is the product of the fitnesses of the independent loci. The sexual species produce genetically varied offspring according to Mendel's laws; the recombination frequency between all loci is 0–5. Members of the asexual species produce offspring that are genetic replicates of themselves. It is important to note that the model represents segregation and independent assortment of genes but not linkage disequilibrium.Computer simulation results were consistent with analytical results, suggesting that inferences can be extrapolated from the analysis without danger of serious error. Both the analysis and simulations reveal a dilemma for the hypothesis that sex is an adaptation to temporal uncertainty; viz., the conditions that are most favorable for sexually are somewhat antithetical (but not prohibitive) to the maintenance of genetic polymorphism in the sexual species whereas sex is useless in a monomorphic population. The dilemma is particularly apparent when only one or a few loci are selected; however, as the number of selected loci increases, the disadvantage in sexuality diminishes. Thus, environmental uncertainty may explain the adaptive significance of sex provided many loci are selected in the prescribed manner.     

Regulated growth in random environments

Summary Conditions for extinction, convergence to a stationary distribution and attaining a carrying capacity are given for stochastic versions of the logistic growth process.     

Population growth in random environments

This paper reviews some recent advances in single population stochastic differential equation growth models. They are a natural way to model population growth in a randomly varying environment. The question of which calculus, Itô or Stratonovich, is preferable is addressed. The two calculi coincide when the noise term is linear, if we take into account the differences in the interpretation of the parameters. This clarifies, among other things, the controversy on the theory of niche limiting similarity proposed by May and MacArthur. The effects of correlations in the environmental fluctuations and statistical methods for estimating parameters and for prediction based on a single population trajectory are mentioned. Applications to fisheries, wildlife management and particularly to environmental impact assessment are now becoming possible and are proposed in this paper.     

Cycle distributions in random nets

Characteristics of random nets are derived from assumptions concerning the distribution of connections. The single aggregate of neurons with random connections without branching and two parallel chains with normal distribution of connections are considered. The cycle saturation is derived for each type of net, that is, the fraction of neurons which are members of cycles. It is shown that in the single aggregate with random connections, the cycle saturation varies inversely as the square root of the number of neurons; in the dense two-chain net it varies inversely as the square root of the neuron density and inversely on the square root of the standard deviation of the normal distribution.     

Persistence of structured populations in random environments

Environmental fluctuations often have different impacts on individuals that differ in size, age, or spatial location. To understand how population structure, environmental fluctuations, and density-dependent interactions influence population dynamics, we provide a general theory for persistence for density-dependent matrix models in random environments. For populations with compensating density dependence, exhibiting “bounded” dynamics, and living in a stationary environment, we show that persistence is determined by the stochastic growth rate (alternatively, dominant Lyapunov exponent) when the population is rare. If this stochastic growth rate is negative, then the total population abundance goes to zero with probability one. If this stochastic growth rate is positive, there is a unique positive stationary distribution. Provided there are initially some individuals in the population, the population converges in distribution to this stationary distribution and the empirical measures almost surely converge to the distribution of the stationary distribution. For models with overcompensating density-dependence, weaker results are proven. Methods to estimate stochastic growth rates are presented. To illustrate the utility of these results, applications to unstructured, spatially structured, and stage-structured population models are given. For instance, we show that diffusively coupled sink populations can persist provided that within patch fitness is sufficiently variable in time but not strongly correlated across space.     

Extinction and exponential growth in random environments.

The influence of randomly varying environments on unrestricted population growth and extinction is analyzed by means of branching processes with random environments (BPRE). A main theme is the interplay between environmental and sampling (or “demographic”) variability. If the two sources of variationg are of comparable magnitude, the environmental variation will dominate except as regards the event of extinction.A diffusion approximation of BPRE is proposed to study the situation of a large population with small environmental variance and mean offspring size near one.Comments on the ecological literature as well as on the relation of the results to previous work involving stochastic differential equations are also given.     

An uncertain life: demography in random environments

This paper concisely reviews the demography of populations with random vital rates, highlights examples and techniques which yield insight into population dynamics, summarizes the state of significant applications of the theory, and points to open problems. The central picture in this theory is of a time-varying but statistically stationary equilibrium for population, sharply distinct from the notions of classical demography. The deepest biological insights from the theory reveal the temporal structure of life histories to be a rich arena for natural selection.     

Invariant probabilities for systems in a random environment—With applications to the Brusselator

In this paper we are concerned with problems of the long-term behavior for nonlinear systems in random environment. The general model is assumed to be given by an ordinary differential equation with random parameters or random input. The disturbance process can be taken from a fairly general class of Markov processes having a bounded state space. In terms of the system’s dynamics we give sufficient conditions for the existence and uniqueness of invariant probabilities. Finally, we apply these results to the two-dimensional biochemical model which is known as the Brusselator. This work is part of a research project supported by the ‘Stiftung Volkswagenwerk’.     

Deterministic genetic models in varying environments

Summary J. B. S. Haldane and S. D. Jayakar [J. Genet. 58, 237–242 (1963)] argue that, when genotype fitnesses fluctuate from generation to generation, if the geometric and arithmetic means of the fitnesses satisfy certain inequalities, there will be a protected polymorphism. Their assertions are biologically interesting, but their mathematical analysis is not sufficient to support their conclusions. We present a firm mathematical analysis and several examples that demonstrate the need for stronger hypotheses and, in some cases, weaker conclusions.Journal Paper No. J-10136 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa. Project 1669. Partial support by National Institutes of Health, Grant GM 13827     

Predator-prey models in periodically fluctuating environments

A class of ordinary or integrodifferential equations describing predator-prey dynamics is considered under the assumption that the coefficients are periodic functions of time. This class is characterized by the logistic behaviour of the prey in the absence of predators and it includes the Leslie model. We show that there exists a periodic solution provided that the average of the predator's intrinsic rate of increase is greater than a critical value. We use well-known results in bifurcation theory for nonlinear eigenvalue problems, as well as an extension to the case of non-globally defined operators of some recent results on the global nature of branches of solutions.     

Prey-predator models in spatially heterogeneous environments

The effects of environmental heterogeneity on models of prey-predator systems are investigated. Refuge behaviour is found in a continuous gradually varying environment. In this situation we do not necessarily get oscillating population cycles. The stabilizing effect observed depends on environmental variation and is not produced by diffusion alone. Our conclusions are fairly independent of the details of the model.     

RMBNToolbox: random models for biochemical networks

Background  

There is an increasing interest to model biochemical and cell biological networks, as well as to the computational analysis of these models. The development of analysis methodologies and related software is rapid in the field. However, the number of available models is still relatively small and the model sizes remain limited. The lack of kinetic information is usually the limiting factor for the construction of detailed simulation models.     

Predictive ability of genomic selection models in a multi-population perennial ryegrass training set using genotyping-by-sequencing

Key message

Genomic prediction models for multi-year dry matter yield, via genotyping-by-sequencing in a composite training set, demonstrate potential for genetic gain improvement through within-half sibling family selection.

Abstract

Perennial ryegrass (Lolium perenne L.) is a key source of nutrition for ruminant livestock in temperate environments worldwide. Higher seasonal and annual yield of herbage dry matter (DMY) is a principal breeding objective but the historical realised rate of genetic gain for DMY is modest. Genomic selection was investigated as a tool to enhance the rate of genetic gain. Genotyping-by-sequencing (GBS) was undertaken in a multi-population (MP) training set of five populations, phenotyped as half-sibling (HS) families in five environments over 2 years for mean herbage accumulation (HA), a measure of DMY potential. GBS using the ApeKI enzyme yielded 1.02 million single-nucleotide polymorphism (SNP) markers from a training set of n = 517. MP-based genomic prediction models for HA were effective in all five populations, cross-validation-predictive ability (PA) ranging from 0.07 to 0.43, by trait and target population, and 0.40–0.52 for days-to-heading. Best linear unbiased predictor (BLUP)-based prediction methods, including GBLUP with either a standard or a recently developed (KGD) relatedness estimation, were marginally superior or equal to ridge regression and random forest computational approaches. PA was principally an outcome of SNP modelling genetic relationships between training and validation sets, which may limit application for long-term genomic selection, due to PA decay. However, simulation using data from the training experiment indicated a twofold increase in genetic gain for HA, when applying a prediction model with moderate PA in a single selection cycle, by combining among-HS family selection, based on phenotype, with within-HS family selection using genomic prediction.