A first principles derivation of animal group size distributions

Several empirical studies have shown that the animal group size distribution of many species can be well fit by power laws with exponential truncation. A striking empirical result due to Niwa is that the exponent in these power laws is one and the truncation is determined by the average group size experienced by an individual. This distribution is known as the logarithmic distribution. In this paper we provide first principles derivation of the logarithmic distribution and other truncated power laws using a site-based merge and split framework. In particular, we investigate two such models. Firstly, we look at a model in which groups merge whenever they meet but split with a constant probability per time step. This generates a distribution similar, but not identical to the logarithmic distribution. Secondly, we propose a model, based on preferential attachment, that produces the logarithmic distribution exactly. Our derivation helps explain why logarithmic distributions are so widely observed in nature. The derivation also allows us to link splitting and joining behavior to the exponent and truncation parameters in power laws.     

Population dynamics in variable environments I. Long-run growth rates and extinction

A model of population growth is studied in which the Leslie matrix for each time interval is chosen according to a Markov process. It is shown analytically that the distribution of total population number is lognormal at long times. Measures of population growth are compared and it is shown that a mean logarithmic growth rate and a logarithmic variance effectively describe growth and extinction at long times. Numerical simulations are used to explore the convergence to lognormality and the effects of environmental variance and autocorrelation. The results given apply to other geometric growth models which involve nonnegative growth matrices.     

Contribution to the probabilistic theory of neural nets: I. Randomization of refractory periods and of stimulus intervals

Aggregates of neurons are considered in which the frequency of occurrence of neurons with a specified value of the refractory period follows certain probability distributions. Input-output functions are derived for such aggregates. In particular, if input and output intensities are defined in terms of stimulus frequencies and firing frequencies per neuron respectively, it is shown that a rectangular distribution of refractory periods leads to a logarithmic input-output curve. If input and output are defined in terms of the total number of stimuli and firings in the aggregate, it is shown how the “mobilization” picture leads to the logarithmic input-output curve. By randomizing the intervals between stimuli received by a single neuron and by introducing an inhibitory neuron a very simple “filter net” can be constructed whose output will be sensitive to a particular range of the input, and this range can be made arbitrarily small.     

Some quantitative relationships between leaf area index and canopy nitrogen content and distribution

In a previous study (Yin et al. 2000. Annals of Botany 85: 579-585), a generic logarithmic equation for leaf area index (L) in relation to canopy nitrogen content (N) was developed: L=(1/ktn)1n(1+ktnN/nb). The equation has two parameters: the minimum leaf nitrogen required to support photosynthesis (nb), and the leaf nitrogen extinction coefficient (ktn). Relative to nb, there is less information in the literature regarding the variation of ktn. We therefore derived an equation to theoretically estimate the value of ktn. The predicted profile of leaf nitrogen in a canopy using this theoretically estimated value of ktn is slightly more uniform than the profile predicted by the optimum nitrogen distribution that maximizes canopy photosynthesis. Relative to the optimum profile, the predicted profile is somewhat closer to the observed one. Based on the L-N logarithmic equation and the theoretical ktn value, we further quantified early leaf area development of a canopy in relation to nitrogen using simulation analysis. In general, there are two types of relations between L and N, which hold for canopies at different developmental phases. For a fully developed canopy where the lowest leaves are senescing due to nitrogen shortage, the relationship between L and N is described well by the logarithmic model above. For a young, unclosed canopy (i.e. L < 1.0), the relation between L and N is nearly linear. This linearity is virtually the special case of the logarithmic model when applied to a young canopy where its total nitrogen content approaches zero and the amount of nitrogen in its lowest leaves is well above nb. The expected patterns of the L-N relationship are discussed for the phase of transition from young to fully developed canopies.     

Virtual power based algorithm for decoupling large motions from infinitesimal strains: application to shoulder joint biomechanics

New trends of numerical models of human joints require more and more computation of both large amplitude joint motions and fine bone stress distribution. Together, these problems are difficult to solve and very CPU time consuming. The goal of this study is to develop a new method to diminish the calculation time for this kind of problems which include calculation of large amplitude motions and infinitesimal strains. Based on the Principle of Virtual Power, the present method decouples the problem into two parts. First, rigid body motion is calculated. The bone micro-deformations are then calculated in a second part by using the results of rigid body motions as boundary conditions. A finite element model of the shoulder was used to test this decoupling technique. The model was designed to determine the influence of humeral head shape on stress distribution in the scapula for different physiological motions of the joint. Two versions of the model were developed: a first version completely deformable and a second version based on the developed decoupling method. It was shown that biomechanical variables, as mean pressure and von Mises stress, calculated with the two versions were sensibly the same. On the other hand, CPU time needed for calculating with the new decoupled technique was more than 6 times less than with the completely deformable model.     

Graded and discrete receptor potentials in the compound eye of the Australian Bulldog-ant (Myrmecia gulosa)

1. Graded and discrete receptor potentials are recorded from the visual cells of the Australian Bulldog-ant. The intensity dependence of the graded responses is described by a new formula [Eq. (3)]. While the frequency of the discrete potentials in relation to the number of light quanta fits best a Poisson distribution, the graded potentials are best described by a logarithmic Gaussian distribution. 2. It is shown that the non-linear summation of single bumps and the reduction of the bump amplitude lead to a logarithmic intensity dependence. 3. The frequency spectrum of single bumps is measured with a Fast-Fourier-Analysis. It is observed that the harmonic frequencies have a negative slope around 12 dB/octave. 4. A difference is found in the higher harmonics of bumps generated at lower light intensities from those generated at higher light intensities. It is shown that this difference becomes more obvious if the bumps are further divided into short and longer latency groups. 5. From these results, it is concluded, that there is a mutual influence between neighbouring visual cells. Using this influence as a basis, a model of the low electric coupling between the cells is discussed.     

Virtual Power Based Algorithm for Decoupling Large Motions from Infinitesimal Strains: Application to Shoulder Joint Biomechanics

New trends of numerical models of human joints require more and more computation of both large amplitude joint motions and fine bone stress distribution. Together, these problems are difficult to solve and very CPU time consuming. The goal of this study is to develop a new method to diminish the calculation time for this kind of problems which include calculation of large amplitude motions and infinitesimal strains. Based on the Principle of Virtual Power, the present method decouples the problem into two parts. First, rigid body motion is calculated. The bone micro-deformations are then calculated in a second part by using the results of rigid body motions as boundary conditions. A finite element model of the shoulder was used to test this decoupling technique. The model was designed to determine the influence of humeral head shape on stress distribution in the scapula for different physiological motions of the joint. Two versions of the model were developed: a first version completely deformable and a second version based on the developed decoupling method. It was shown that biomechanical variables, as mean pressure and von Mises stress, calculated with the two versions were sensibly the same. On the other hand, CPU time needed for calculating with the new decoupled technique was more than 6 times less than with the completely deformable model.     

Heterogeneous communities with lognormal species abundance distribution: Species-area curves and sustainability

Heterogeneous species abundance models are models in which the dynamics differ between species, described by variation among parameters defining the dynamics. Using a dynamic and heterogeneous species abundance model generating the lognormal species abundance distribution it is first shown that different degrees of heterogeneity may result in equivalent species abundance distributions. An alternative to Preston's canonical lognormal model is defined by assuming that reduction in resources, for example reduction in available area, increases the density regulation of each species. This leads to species-individual curves and species-area curves that are approximately linear in a double logarithmic plot. Preston's canonical parameter gamma varies little along these curves and takes values in the neighborhood of one. Quite remarkably, the curves, which define the sensitivity of the community to area reductions, are independent of the heterogeneity among species for this model. As a consequence, the curves can be estimated from a single sample from the community using the Poisson lognormal distribution. It is shown how to perform sensitivity analysis with respect to over-dispersion in sampling relative to the Poisson distribution as well as sampling intensity, that is, the fraction of the community sampled. The method is exemplified by analyzing three simulated data sets.     

A canonical model for production and distribution of root mass in space and time

A mathematical model is presented for the production and distribution of plant root mass in a substrate in space and time. This model is intended to be used for the interpretation of data on root density profiles. It is assumed that a pulse of root apices appears at the surface of the substrate at zero time, emerging from seeds or stems, and that these apices migrate downwards in the substrate, leaving trails of root mass as they go, giving birth to daughter apices, and dying by terminal differentiation. The model has at least 3 parameters, representing, respectively, initial root generating capacity of root axes at zero time, net rate of root apex reproduction (or extinction), and rate of downwards migration of root apices. Two versions of the model are explored, a nondiffusive version and a diffusive version in which there is an additional parameter representing diffusivity of root apices. It is shown that both versions of this theoretical mechanistic model predict that the profile of root density with depth tends to a steady-state exponential form, parameterised by a density of root mass at zero depth and a logarithmic rate of change of root mass with respect to depth. This form has previously appeared in the literature as an empirical model. Data analyses are performed, using this model, on a set of data in which cultures of plants were subjected to treatment with auxin or by genetic manipulation, and the effects of the various treatments on the parameters of the model are described. Interpretations of the experimental results are proposed in the light of the theoretical results derived in this paper.     

On measures of association among genetic variables

Systems involving many variables are important in population and quantitative genetics, for example, in multi-trait prediction of breeding values and in exploration of multi-locus associations. We studied departures of the joint distribution of sets of genetic variables from independence. New measures of association based on notions of statistical distance between distributions are presented. These are more general than correlations, which are pairwise measures, and lack a clear interpretation beyond the bivariate normal distribution. Our measures are based on logarithmic (Kullback-Leibler) and on relative 'distances' between distributions. Indexes of association are developed and illustrated for quantitative genetics settings in which the joint distribution of the variables is either multivariate normal or multivariate-t, and we show how the indexes can be used to study linkage disequilibrium in a two-locus system with multiple alleles and present applications to systems of correlated beta distributions. Two multivariate beta and multivariate beta-binomial processes are examined, and new distributions are introduced: the GMS-Sarmanov multivariate beta and its beta-binomial counterpart.     

Measurement scale in maximum entropy models of species abundance

The consistency of the species abundance distribution across diverse communities has attracted widespread attention. In this paper, I argue that the consistency of pattern arises because diverse ecological mechanisms share a common symmetry with regard to measurement scale. By symmetry, I mean that different ecological processes preserve the same measure of information and lose all other information in the aggregation of various perturbations. I frame these explanations of symmetry, measurement, and aggregation in terms of a recently developed extension to the theory of maximum entropy. I show that the natural measurement scale for the species abundance distribution is log-linear: the information in observations at small population sizes scales logarithmically and, as population size increases, the scaling of information grades from logarithmic to linear. Such log-linear scaling leads naturally to a gamma distribution for species abundance, which matches well with the observed patterns. Much of the variation between samples can be explained by the magnitude at which the measurement scale grades from logarithmic to linear. This measurement approach can be applied to the similar problem of allelic diversity in population genetics and to a wide variety of other patterns in biology.     

Geometric and probabilistic stability criteria for delay systems.

A new approach to the study of the stability of delay systems is developed. The method is applicable to biological control systems and other systems where little information about time delays is available. The view proposed is that stability information can be deduced from the statistical properties of the probability distribution that encodes the structure of the time delay. The main statistical variables used are the usual expectation parameter E and a modified variance, called relative variance and denoted R, that is invariant under time-scale changes. In many cases, the stability of a model improves as R increases while E remains fixed. The statistical approach is shown to be closely related to a geometric method of Walther and Cushing that establishes stability in the case of a convex delay distribution function. In fact, it is shown that convex and concave distributions have R values respectively greater than and less than 1/2. A generalized version of the geometric theory is presented that relaxes the smoothness hypothesis on the density function; this brings it more into correspondence with statistical theory, which applies to general distributions irrespective of their smoothness.     

Migration through surnames in Campobasso Province, Italy

Data on grandparental surnames were obtained from school-children in 22 communes from Campobasso Province, Italy (Molise Region). The distribution of surnames was shown to be almost exactly linear by a log2-log2 transformation, which justified the fitting of the data to Fisher's logarithmic distribution. The values for v were higher among women. When v was standardized to minimize bias due to sample size, the value was one-third the estimate of migration from exogamy data. The higher values of v for females indicate that there is greater mobility of female marriage partners than males.     

Mathematical biology of division of labor between two individuals or two social groups

The concept of the satisfaction function is applied to the situation in which two individuals may each produce two needed objects of satisfaction, or may each produce only one of the objects and then make a partial exchange. It is shown that with a logarithmic satisfaction function there is no advantage in a division of labor, unless such a division materially increases the purely physical efficiency of production. This result appears to be connected with the particular choice of the form of satisfaction function (logarithmic). While the problem has not been solved for other forms, it is made plausible that satisfaction functions which have an asymptote will lead to a different result. Next the case is studied in which division of labor occurs between two groups of individuals. It is shown that in this case the relative sizes of the two groups are determined from considerations of maximum satisfaction. Possible applications to problems of urbanization are suggested.     

The Sampling Distribution of Disease-Associated Alleles

A theory is developed that provides the sampling distribution of low frequency alleles at a single locus under the assumption that each allele is the result of a unique mutation. The numbers of copies of each allele is assumed to follow a linear birth-death process with sampling. If the population is of constant size, standard results from theory of birth-death processes show that the distribution of numbers of copies of each allele is logarithmic and that the joint distribution of numbers of copies of k alleles found in a sample of size n follows the Ewens sampling distribution. If the population from which the sample was obtained was increasing in size, if there are different selective classes of alleles, or if there are differences in penetrance among alleles, the Ewens distribution no longer applies. Likelihood functions for a given set of observations are obtained under different alternative hypotheses. These results are applied to published data from the BRCA1 locus (associated with early onset breast cancer) and the factor VIII locus (associated with hemophilia A) in humans. In both cases, the sampling distribution of alleles allows rejection of the null hypothesis, but relatively small deviations from the null model can account for the data. In particular, roughly the same population growth rate appears consistent with both data sets.     

Severity Factor as an Efficient Control Parameter to Predict Biomass Solubilization and Saccharification During Acidic Hydrolysis of Microalgal Biomass

In this paper, the acidic pretreatment of microalgal biomass is investigated, and the solubilized biomass and hydrolyzed sugars were evaluated. The process is analyzed through the severity factor approach (acidic combined severity factor (ACSF)). A suitable kinetic model is developed and applied, and it is shown that the severity factor theory works. A discussion and comparison are presented with respect to the literature methods, which are mainly related to lignocellulosic biomass. In the case of microalgae, reaction orders for biomass and acid are shown to be the main parameters, and no other assumptions are needed. Two regions of the acidic treatment process have to be evaluated: low and high reactivity regions. Furthermore, a suitable experimental design is required in order to provide an appropriate reaction spectrum to obtain a good estimation of the kinetic parameters. A logarithmic severity factor range (ln ACSF) between 5 and 6 is able to solubilize around 80% of biomass and to hydrolyze more than 90% of sugars present in the biomass.     

林网内蒸发水平分布规律的研究

通过水面蒸发观测资料分析,提出林网内相对蒸发变化可用分段曲线拟合,分为对数区与二次曲线区两部分．对数区相对蒸发变化服从对数规律ＥＡ／Ｅ０＝ａ１＋ｂ１ｌｎｈ,二次曲线区相对蒸发变化服从二次曲线规律Ｅｈ／Ｅ０＝ａ２ｈ２十ｂ２ｈ＋ｃ．在蒸发水平分布特征研究基础上,对林网内平均相对蒸发强度进行了估算,并讨论了保水性能最佳防护林带距间题．     

Cation Transport in Escherichia coli : I. Intracellular Na and K concentrations and net cation movement

Methods have been developed to study the intracellular Na and K concentrations in E. coli, strain K-12. These intracellular cation concentrations have been shown to be functions of the extracellular cation concentrations and the age of the bacterial culture. During the early logarithmic phase of growth, the intracellular K concentration greatly exceeds that of the external medium, whereas the intracellular Na concentration is lower than that of the growth medium. As the age of the culture increases, the intracellular K concentration falls and the intracellular Na concentration rises, changes which are related to the fall in the pH of the medium and to the accumulation of the products of bacterial metabolism. When stationary phase cells, which are rich in Na and poor in K, are resuspended in fresh growth medium, there is a rapid reaccumulation of K and extrusion of Na. These processes represent oppositely directed net ion movements against concentration gradients, and have been shown to be dependent upon the presence of an intact metabolic energy supply.     

On the quasi-stationary distribution of the Ross malaria model.

Approximations are derived for the quasi-stationary distribution of the fully stochastic version of the classical Ross malaria model. The approximations are developed in two stages. In the first stage, the Ross process is approximated with a bivariate Markov chain without an absorbing state. The second stage of the approximation uses ideas from perturbation theory to derive explicit expressions that serve as approximations of the joint stationary distribution of the approximating process. Numerical comparisons are made between the approximations and the quasi-stationary distribution.     

STUDY OF BULL SPERMATOZOA : Quantitative Electron Microscopy

The electron microscope has been used to determine the characteristic dimensions and the distribution of the dry mass in bull spermatozoa. The most important result is that all characteristic data are occurring in logarithmic distributions. Furthermore, no correlation between such parameters as head weight and tail weight or head length and tail length was found. The occurrence in logarithmic distributions and the non-correlation of parts in the assembly of a spermatozoon are considered to reflect significant biologic principles. Methodologically, a new procedure is added to quantitative electron microscopy permitting the recording of the mass cross-section (total mass per unit length) of an object. This approach makes possible determinations of the distribution and the total mass of very long and narrow structures.