Intensity of nitrification in soil as a function of time and soil moisture

Proceeding from three previously derived expressions for the intensity of nitrification in soil as a function of time (logΣN=K.logt+q), as a function of incubation moisture (logΣN=A.pF _i+B), as a function of initial moisture (logΣN=C.pF _v+D), it was shown that the nitrification intensity as a function of time and of moisture can be expressed by the bilinear function log ΣN=a.pF _i.logT+b.pF _i+c.logt+d; as a function of time and of initial moisture by the bilinear function logΣ=N=a.pF _v.logt+b.pF _v+c.logt+d; as a function of initial and incubation moisture by the bilinear function log ΣN=a.pF _ipF _v+b.pF _i+c.pF _v+d. The intensity of nitrification as a function of time, incubation moisture and initial moisture may be expressed by the multilinear function log ΣN=a.pF _i.pF _v.logt+b.pF _i.pF _v+c.pF _i.logt+d.pF _v.logt+e .pF _i+f.pF _v=g.logt+h. This function is valid for all the incubation moistures lying between pF _i 3.0 and 4.0 and for all initial moistures between 3.5 and 5.9 provided that the incubation temperature remains constant.     

The Estimation of a Multivariate Fitness Function from Several Samples Taken from a Population

The situation is considered where the multivariate distribution of certain variables X₁, X₂, …, X_p is changing with time in a population because natural selection related to the X's is taking place. It is assumed that random samples taken from the population at times t₁, t₂, …, t_s are available and it is desirable to estimate the fitness function w_t(x₁, x₂,…,x_p) which shows how the number of individuals with X_i = x_i, i = 1, 2, …, p at time t is related to the number of individuals with the same X values at time zero. Tests for population changes are discussed and indices of the selection on the population dispersion and the population mean are proposed. The situation with a multivariate normal distribution is considered as a special case. A maximum likelihood method that can be applied with any form of population distribution is proposed for estimating w_t. The methods discussed in the paper are illustrated with data on four dimensions of male Egyptian skulls covering a time span from about 4500 B.C. to about 300 A.D. In this case there seems to have been very little selection on the population dispersion but considerable selection on means.     

The extinction of slowly evolving dynamical systems

The time evolution of slowly evolving discrete dynamical systems x _{i + 1} = T(r _i ,x _i ), defined on an interval [0, L], where a parameter r _ichanges slowly with respect to i is considered. For certain transformations T, once r _i reaches a critical value the system faces a non-zero probability of extinction because some x _j [0, L]. Recent ergodic theory results of Ruelle, Pianigiani, and Lasota and Yorke are used to derive a simple expression for the probability of survival of these systems. The extinction process is illustrated with two examples. One is the quadratic map, T(r, x) = rx(1 – x), and the second is a simple model for the growth of a cellular population. The survival statistics for chronic myelogenous leukemia patients are discussed in light of these extinction processes. Two other dynamical processes of biological importance, to which our results are applicable, are mentioned.     

Local consideration of polymorphisms for populations coexisting in stable ecosystems

Summary The stability problem for multiallelic genetic polymorphisms for n populations coexisting in stable ecosystems is considered. Taking into account only density-dependent interactions a generalization of Fisher's theorem is obtained. Specifically, the average fitness of a population must be locally maximized subject to the constraint that the equilibrium population sizes are fixed if the polymorphism is stable. Further, the quasi-equilibrium population sizes N _i ^* corresponding to fixing the genetic structure of all populations in the ecosystem at various values have extrema at the equilibrium point. Such an equilibrium can be a maximum, minimum or saddle point depending upon the type of ecosystem under consideration. A simple test separating these cases on the basis of the so-called ecosystem matrix is suggested. The general equilibrium problem is reformulated as a maximization problem under some restrictions. Conditions under which the maximized function can be expressed as ₌₁ ⁿ N_i are formulated.     

VARIABLE SELECTION ON EUROSTA'S GALL SIZE,I: THE EXTENT AND NATURE OF VARIATION IN PHENOTYPIC SELECTION

Natural fluctuations in environmental conditions are likely to induce variation in the intensity or direction of natural selection. A long-term study of the insect, Eurosta solidaginins Fitch (Diptera; Tephritidae), which induces stem galls on the perennial herb Solidago altissima (Asteraceae) was performed to explore the patterns of variation in phenotypic selection. The intensity of selection imposed by parasitoids and predators on gallmaking larvae, for gall size, was measured across 16 populations over the course of 4 generations, for a total of 64 population-generations. Directional selection was quantified by i, the selection intensity, and variance selection by j‘, a measure of the intensity of selection on phenotypic variance. Size-dependent attack by parasitoids caused upward directional selection (mean i_p = 0.42; SE = 0.023), while size-dependent bird attack favored larvae that induced smaller galls (mean i_b = -0.07; SE = 0.013. The mean net directional selection intensity was 0.35 (SE = 0.030), which indicates that insects inducing larger galls are generally favored by selection. The opposing patterns of size-dependent attack resulted in stabilizing selection in half the population generations, with an overall average. j‘ of -0.11 (SE = 0.078). The magnitude of directional selection was strongly influenced by the population mean gall size and weakly by the optimal gall size. The intensity of variance selection was strongly influenced by the shape of the fitness function, with sigmoidal and Gaussian-like shapes causing greater depletion of phenotypic variance.     

Climate‐driven vital rates do not always mean climate‐driven population

Current climatic changes have increased the need to forecast population responses to climate variability. A common approach to address this question is through models that project current population state using the functional relationship between demographic rates and climatic variables. We argue that this approach can lead to erroneous conclusions when interpopulation dispersal is not considered. We found that immigration can release the population from climate‐driven trajectories even when local vital rates are climate dependent. We illustrated this using individual‐based data on a trans‐equatorial migratory seabird, the Scopoli's shearwater Calonectris diomedea, in which the variation of vital rates has been associated with large‐scale climatic indices. We compared the population annual growth rate λ_i, estimated using local climate‐driven parameters with ρ_i, a population growth rate directly estimated from individual information and that accounts for immigration. While λ_i varied as a function of climatic variables, reflecting the climate‐dependent parameters, ρ_i did not, indicating that dispersal decouples the relationship between population growth and climate variables from that between climatic variables and vital rates. Our results suggest caution when assessing demographic effects of climatic variability especially in open populations for very mobile organisms such as fish, marine mammals, bats, or birds. When a population model cannot be validated or it is not detailed enough, ignoring immigration might lead to misleading climate‐driven projections.     

Random median sampling to enhance the precision of population estimates

We describe the characteristics of a sampling procedure called random median sampling that was proposed to enhance the precision of population estimates. In performing random median sampling, we first select a sampling item at random from the sampling area. We roughly compare the abundance of individuals in the selected item with that of the adjacent two items in order to identify the item that has median abundance, i.e., the item that has the second largest abundance among the three items. We count the number of individuals of the item having the median abundance. This procedure is repeated n times in the sampling area (i = 1, 2, ..., n). Let m _i be the ith median abundance. The estimates of the mean abundance per sampling item and the variance of estimates are given by Σm _i /n and Σ(m _i –Σm _i /n)²/n(n – 1), respectively. This method is a local application of the median ranked set sampling that was proposed by Muttlak (J Appl Stat Sci 6:245–255, 1997). Random median sampling is effective when the correlation coefficient between adjacent items is small. If the correlation coefficient is close to zero, random median sampling reduces the variance of estimates to 45 or 32% of that in simple random sampling when the distribution follows a normal distribution or a Laplace distribution, respectively. The sample size required to achieve a given precision of estimate decreases accordingly. The effectiveness of random median sampling, however, is small if the correlation coefficient is large. The condition in which random median sampling is superior to simple random sampling is also discussed.     

Effect of the population heterogeneity on growth behavior and its estimation

Different types of the Logistic model are constructed based on a simple assumption that the microbial populations are all composed of homogeneous members and consequently, the condition of design for the initial value of these models has to be rather limited in the case of N(t ₀)=N ₀. Therefore, these models cannot distinguish the dynamic behavior of the populations possessing the same N ₀ from heterogeneous phases. In fact, only a certain ratio of the cells in a population is dividing at any moment during growth progress, termed as θ, and thus, dN / dt not only depends on N, but also on θ. So θ is a necessary element for the condition design of the initial value. Unfortunately, this idea has long been neglected in widely used growth models. However, combining together the two factors (N ₀ and θ) into the initial value often leads to the complexity in the mathematical solution. This difficulty can be overcome by using instantaneous rates (V _inst) to express growth progress. Previous studies in our laboratory suggested that the V _inst curve of the bacterial populations all showed a Guassian function shape and thus, the different growth phases can be reasonably distinguished. In the present study, the Gaussian distribution function was transformed approximately into an analytical form ( Y_i = ae^{[ - 0.5( \fracx_i - x₀ b )2 ]} Y_i = \alpha e^{\left[ { - 0.5\left( {\frac{{x_i - x_0 }}{b}} \right)^2 } \right]} ) that can be conveniently used to evaluate the growth parameters and in this way the intrinsic growth behavior of a bacterial species growing in heterogeneous phases can be estimated. In addition, a new method has been proposed, in this case, the lag period and the double time for a bacterial population can also be reasonably evaluated. This approach proposed could thus be expected to reveal important insight of bacterial population growth. Some aspects in modeling population growth are also discussed.     

Some Simple Statistics to Test Against Ordered Alternatives in Complete Block Designs

Consider testing the hypothesis of no treatment effects against a postulated ranking of the m treatments, given data from n Complete Blocks. A suitable test statistic is the weighted average rank correlation w = σb_QiC_i where C_i is the correlation between the postulated ranking and the ranking observed within the ith block, Q_i is the rank of the ith block with respect to credibility, and the b_i's are weights such that 0 ≦ b¹ ≦ … ≦ bn. In this paper we introduce some simple statistics: the first extends the signed-rank statistic to m ≦ 3, the second uses a simple measure of correlation based on the antirank, and the third a statistic based on Spearman's footrule. Tables for critical values are provided and the normal approximation is investigated.     

Sampling error as a misleading artifact in “key factor analysis”

The influences of sampling error in key factor analysis are investigated statistically. The error involved in the data distorts the results in various misleading ways. In the course of detecting key factors by correlation analysis, the distortion arises in the following two ways: (1) the contributions made by the first and the last components of population trend index (log I or K) to the total variation are overrated as compared with the others; and (2) spurious negative correlation arises between successive two components. The risk of misinterpretation due to such disturbance is usually increased further if the error is concentrated on any particular developmental stages. In the tests to detect density-dependence by using regression analysis, the error consistently acts as if it were a density-dependent factor: under the effect of sampling error, the slope b for the regression of log N_i+1 on log N_i, for example, is expected to become<1 even where there is no density-dependent factor at all. A set of formulas are derived which may serve to check and correct these misleading distortions caused by the error. It is also shown that such undesirable influences can be avoided, at least to a considerable extent, if appropriate sampling plans are adopted for the study. The validity of key factor analysis is discussed in reference to this and some related problems.     

Galerkin-Ritz procedures for approximate solutions to systems of reaction-diffusion equations

For any essentially nonlinear system of reaction-diffusion equations of the generic form ∂c_i/∂t=D_i∇²c_i+Q_i(c,x,t) supplemented with Robin type boundary conditions over the surface of a closed bounded three-dimensional region, it is demonstrated that all solutions for the concentration distributionn-tuple function c=(c ₁(x,t),...,c _n (x,t)) satisfy a differential variational condition. Approximate solutions to the reaction-diffusion intial-value boundary-value problem are obtainable by employing this variational condition in conjunction with a Galerkin-Ritz procedure. It is shown that the dynamical evolution from a prescribed initial concentrationn-tuple function to a final steady-state solution can be determined to desired accuracy by such an approximation method. The variational condition also admits a systematic Galerkin-Ritz procedure for obtaining approximate solutions to the multi-equation elliptic boundary-value problem for steady-state distributions c=−c(x). Other systems of phenomenological (non-Lagrangian) field equations can be treated by Galerkin-Ritz procedures based on analogues of the differential variational condition presented here. The method is applied to derive approximate nonconstant steady-state solutions for ann-species symbiosis model.     

The rate of convergence of a generalized stable population

In an age-structured population that grows exponentially, each age groupP _i(t) at periodt is asymptotically equivalent tox ₀ ^t for some positive number x₀. In this paper we show that the speed at which the ith age group reaches its exponential state of equilibrium can be measured by the rate at which the ratio v_i(t)=P_i(t)/p_i(t–1) converges tox ₀. The age specific rate of convergence is determined by considering a quantityr satisfyingv _i(t)-x ₀ ¦ r ^t whent is large;R _i=Infr (over all initial populations,r satisfying the above inequality) is the R-factor used in numerical analysis to measure the rate at which the sequencev _i (t) converges tox ₀;S _i =- In R_i is then defined as the rate of convergence to stability of the ith age group. The case of constant net maternity rates is studied in detail; in this contextS ₀ is compared to the population entropyH, which was proposed by Tuljapurkar (1982) as a measure of the rate of convergence to stability.     

On a Class of Almost Unbiased Ratio Estimators

A class of almost unbiased ratio estimators for population mean σ is derived by weighting sample σ = (1/n) σ y_i, ratio estimators σ and an estimator, σ (y_i/x_i). It is shown that NIETO DE PASCUAL (1961) estimator is a particular member of the class and an optimum estimator in the class (in the minimum variance sense) is identified. The results are illustrated through two numerical examples.     

Generalizing Double and Triple Sampling for Repeated Surveys and Partial Verification

Population composition is often estimated by double sampling in which the value of a covariate is noted on each of a large number of randomly selected units and the value of the covariate and the exact class to which the unit belongs is noted for a smaller sample. The cross‐classified sample can be used to estimate the classification rates and these, in turn, can be used in conjunction with the estimated distribution of the covariate to obtain an improved estimate of the population composition over that obtained by direct observation of the identity of the individuals in a small sample. There are two approaches to this problem characterized by the way in which the classification rates are defined. The simplest approach uses estimates of the probability P(i | j) that the unit is actually in class i given that the covariate is in class j. The more complicated approach uses estimates of the probability Pi | j) that the covariate falls in class j given that the unit is actually in class i. The latter approach involves estimating more parameters than the former but avoids the necessity for the two samples to be drawn from the same population. We show the two approaches can be combined when there are multiple surveys. For example, one might conduct a disease survey for several years; in each year the accurate and/or error‐prone techniques may be applied to samples. The sensitivities and specificities of the error‐prone test are assumed constant across surveys. Generalizations allow for more than one error‐prone classifier and partial verification (estimation of misclassification rates by application of the accurate technique to fixed subsamples from each error‐prone category). The general approach is illustrated by considering a repeated survey for malaria.     

A model of DNA sequence evolution

Statistical studies of gene populations on the purine/pyrimidine alphabet have shown that the mean occurrence probability of thei-motif YRY(N) _i YRY (R=purine, Y=pyrimidine, N=R or Y) is not uniform by varyingi in the range [1,99], but presents a maximum ati=6 in the following populations: protein coding genes of eukaryotes, prokaryotes, chloroplasts and mitrochondria, and also viral introns, ribosomal RNA genes and transfer RNA genes (Arquès and Michel, 1987b,J. theor. Biol. 128, 457–461). From the “universality” of this observation, we suggested that the oligonucleotide YRY(N)₆ is a primitive one and that it has a central function in DNA sequence evolution (Arquès and Michel, 1987b,J. theor. Biol. 128, 457–461). Following this idea, we introduce a concept of a model of DNA sequence evolution which will be validated according to a shema presented in three parts. In the first part, using the last version of the gene database, the YRY(N)₆YRY preferential occurrence (maximum ati=6) is confirmed for the populations mentioned above and is extended to some newly analysed populations: chloroplast introns, chloroplast 5′ regions, mitochondrial 5′ regions and small nuclear RNA genes. On the other hand, the YRY(N)₆YRY preferential occurrence and periodicities are used in order to classify 18 gene populations. In the second part, we will demonstrate that several statistical features characterizing different gene populations (in particular the YRY(N)₆YRY preferential occurrence and the periodicities) can be retrieved from a simple Markov model based on the mixing of the two oligonucleotides YRY(N)₆ and YRY(N)₃ and based on the percentages of RYR and YRY in the unspecified trinucleotides (N)₃ of YRY(N)₆ and YRY(N)₃. Several properties are identified and prove in particular that the oligonucleotide mixing is an independent process and that several different features are functions of a unique parameter. In the third part, the return of the model to the reality shows a strong correlation between reality and simulation concerning the presence of large alternating purine/pyrimidine stretches and of periodicities. It also contributes to a greater understanding of biological reality, e.g. the presence or the absence of large alternating purine/pyrimidine stretches can be explained as being a simple consequence of the mixing of two particular oligonucleotides. Finally, we believe that such an approach is the first step toward a unified model of DNA sequence evolution allowing the molecular understanding of both the origin of life and the actual biological reality.     

Optimal Fixing of the Bandwidth-Parameter for the Empirical Regression1,2

The estimator ?₀(x) of the regression r(x) = E (Y | × = x) from measured points (x_i, y_i), i = 1(1) n, of a continuous two-dimensional random variable (X, Y) with unknown continuous density function f(x, y) and with moments up to the second order can be made with the help of a density estimation f?₀(x, y) (see e.g. SCHMERLING and PEIL, 1980). Here f?₀(x, y) still contains free parameters (so-called band-width-parameters), the values of which have to be optimally fixed in the concrete case. This fixing can be done by using a modification of the maximum-likelihood principle including jackknife techniques. The parameter values can be also found from the estimators for r(x). Here the cross-validation principle can be applied. Some numerical aspects of these possibilities for optimally fixing the bandwidth-parameter are discussed by means of examples. If ?₀(x) is used as a smoothing operator for time series the optimal choice of the parameter values is dependent on the purpose of application of the smoothed time series. The fixing will then be done by considering the so-called filter-characteristic of ?C₀(x).     

Genetic evaluation with autosomal and X-chromosomal inheritance

Summary At present, genetic evaluation in livestock using best linear unbiased prediction (BLUP) assumes autosomal inheritance. There is evidence, however, of X-chromosomal inheritance for some traits of economic importance. BLUP can accommodate models that include X-chromosomal in addition to autosomal inheritance. To obtain BLUP with autosomal and X-chromosomal additive inheritance for a population in which allelic frequency is equal in the sexes, and that is in gametic equilibrium, we write y _i = x_i + a_i + s_i + e_i, where y _i is the phenotypic value for individual i, x_i, is a vector of constants relating y _i to fixed effects, is a vector of fixed effects, a _i is the additive genetic effect for autosomal loci, S _i is the additive genetic effect for X-chromosomal loci, and e _i is random error. The covariance matrix of a _i's is A _A ² , where A is the matrix of twice the co-ancestries between relatives for autosomal loci, and _A ² is the variance of additive genetic effects for autosomal loci. The covariance matrix of s _i's is S _F ² , where S is a matrix of functions of co-ancestries between relatives for X-chromosomal loci and _F ² is the variance of additive genetic effects for X-chromosomal loci for noninbred females. Given the covariance matrices of random effects a _i, s_i, and e _i, BLUPs of autosomal and of X-chromosomal additive effects can be obtained using mixed model equations. Recursive rules to construct S and an efficient algorithm to compute its inverse are given.Dedicated to the memory of Dr. C. R. Henderson, whose encouraging comments stimulated the research in this paper. Supported in part by the Illinois Agricultural Experiment Station, Hatch Project 35-0367, Estimation of Genetic Parameters.     

Estimation of a Conditional Mean in a Linear Regression Model

Consider the two linear regression models of Y_ij on X_ij, namely Y_ij = β_io + β_il X_ij + ε_ij,j = 1,2,…,n_i, i = 1,2, where ε_ij are assumed to be normally distributed with zero mean and common unknown variance σ². The estimated value of a mean of Y₁ for a given value of X₁ is made to depend on a preliminary test of significance of the hypothesis β₁₁ = β₂₁. The bias and the mean square error of the estimator for the conditional mean of Y₁ are given. The relative efficiency of the estimator to the usual estimator is computed and is used to determine a proper choice of the significance level of the preliminary test.     

Trumpeter Swan Abundance and Growth Rates in Yellowstone National Park

ABSTRACT Decreasing abundance of resident, nonmigratory trumpeter swans (Cygnus buccinator) in Yellowstone National Park (YNP), USA, raised concern that this population, which helped facilitate the restoration of the species across North America, may disappear. We quantified trends in abundance of resident and migratory trumpeter swans in YNP from 1967 to 2007 and investigated the potential mechanisms for declining population trends, including cessation of the supplemental feeding program and relocation programs outside of YNP, density dependence, and annual variations in environmental conditions. Estimated abundance of resident trumpeter swans in YNP ranged from 59 individuals in 1968 to 10 individuals in 2007. Using log-linear modeling, the best approximating model chosen from an a priori set of competing models estimated the annual growth rate (r) of resident swans from 1967 to 2007 was −0.036 (95% CI =−0.042 to −0.030, Akaike wt [w_i] = 0.44). A competing model provided evidence that decreases in abundance became more dramatic after supplemental feeding of grain outside of YNP was terminated in winter 1992–1993 (r̂_1967–1992 = −0.027, 95% CI = −0.039 to −0.015; r̂_1993–2007 = −0.053, 95% CI = −0.029 to −0.080; w_i = 0.42). There was little evidence of density-dependent effects on the resident population growth rates (βYNP_pop = 0.006, 95% CI = −0.017 to 0.007), but rates were lower following severe winters, wetter springs, and warmer summers. Our results indicate that the YNP population of trumpeter swans is decreasing and may act as a sink to surrounding populations. Thus, population levels of YNP trumpeter swans may depend on management outside the Park and we recommend the National Park Service collaborate with surrounding agencies in managing trumpeter swans throughout the Tri-state region where more productive habitats may exist.     

The Point Evaluation Method for Approximating Function Means,Variances and Covariances

A perennial problem in statistics is the determination of biases, variances and covariances for functions of random variables X₁, X₂, …, X_n which themselves have a known distribution. A common approach is through equations based upon Taylor series approximations but a “point evaluation” method may sometimes be a useful alternative. This involves approximating the multivariate distribution of the X variables by the 2ⁿ points given by X₁=μ₁±₁, X₂ = μ₂ ±₂, …, X_n = = μ_n μ_n, where μ_i is the mean and σ_i the standard deviation of Xi, with appropriate point weights. An advantage over the Taylor series approach is that function derivatives do not have to be explicitely calculated. The point evaluation method is particularly useful in cases where the X variables are uncorrelated. Then the evaluation of the 2ⁿ points can be replaced by the evaluation of 2n points. The point evaluation method is illustrated with powers of a normally distributed variable, and with estimation of gene frequencies from ABO blood group frequencies.