Moment expansions in spatial ecological models and moment closure through Gaussian approximation

We describe the dynamics of competing species in terms of interactions between spatial moments. We close the moment hierarchy by employing a Gaussian approximation which assumes that fluctuations are independent and distributed normally about the mean values. The Gaussian approximation provides the lowest-order systematic correction to the mean-field approximation by incorporating the effect of fluctuations. When there are no fluctuations in the system, the mean equations agree with the Gaussian approximation as the fluctuations are weak. As the fluctuations gain strength, they influence the mean quantities and hence the Gaussian approximation departs from the mean-field approximation. At large fluctuation levels, the Gaussian approximation breaks down, as may be explained by the bimodality and skewness of the fluctuation distribution of the partial differential equation.     

Spatial order as a source of kinetic cooperativity in organized bound enzyme systems.

When enzyme molecules are distributed within a negatively charged matrix, the kinetics of the conversion of a negatively charged substrate into a product depends on the organization of fixed charges and bound enzyme molecules. Organization is taken to mean the existence of macroscopic heterogeneity in the distribution of fixed charge density, or of bound enzyme density, or of both. The degree of organization is quantitatively expressed by the monovariate moments of charge and enzyme distributions as well as by the bivariate moments of these two distributions. The overall reaction rate of the bound enzyme system may be expressed in terms of the monovariate moments of the charge density and of the bivariate moments of charge and enzyme densities. The monovariate moments of enzyme density do not affect the reaction rate. With respect to the situation where the fixed charges and enzyme molecules are randomly distributed in the matrix, the molecular organization, as expressed by these two types of moments, generates an increase or decrease of the overall reaction rate as well as a cooperativity of the kinetic response of the system. Thus both the alteration of the rate and the modulation of cooperativity are the consequence of a spatial organization of charges with respect to the enzyme molecules. The rate equations have been derived for different types of organization of fixed charges and enzyme molecules, namely, clustered charges and homogeneously distributed enzyme molecules, clustered enzyme molecules and homogeneously distributed charges, clusters of charges and clusters of enzymes that partly overlap, and clusters of enzymes and clusters of charges that are exactly superimposed. Computer simulations of these equations show how spatial molecular organization may modulate the overall reaction rate.     

Turbulent transport of suspended particles and dispersing benthic organisms: the hitting-distance problem for the local exchange model

The local exchange model developed by McNair et al. (1997) provides a stochastic diffusion approximation to the random-like motion of fine particles suspended in turbulent water. Based on this model, McNair (2000) derived equations governing the probability distribution and moments of the hitting time, which is the time until a particle hits the bottom for the first time from a given initial elevation. In the present paper, we derive the corresponding equations for the probability distribution and moments of the hitting distance, which is the longitudinal distance a particle has traveled when it hits the bottom for the first time. We study the dependence of the distribution and moments on a particle's initial elevation and on two dimensionless parameters: an inverse Reynolds number M (a measure of the importance of viscous mixing compared to turbulent mixing of water) and the Rouse number ?(a measure of the importance of deterministic gravitational sinking compared to stochastic turbulent mixing in governing the vertical motion of a particle). We also compute predicted hitting-distance distributions for two published data sets. The results show that for fine particles suspended in moderately to highly turbulent water, the hitting-distance distribution is strongly skewed to the right, with mode相似文献   

The relationship between the transition matrix model and the diffusion model

The diffusion equation model and the Lefkovitch matrix model have been employed independently in plant population ecology in order to analyze the dynamics of growth and size structure. The two models describe the dynamics of size structure in biological populations, and thus there must be some relationship between them. In the present paper, we examine the theoretical relationship between these two models. We demonstrate, on a certain assumption, that the one-step Lefkovitch matrix model corresponds to a difference equation of the diffusion equation and that the two- and three-step Lefkovitch matrix model correspond to difference equations of the 4th- and 6th-order Kramers-Moyal expansions, respectively. It is also shown that 2n moments (the first to the 2n-th moments) of growth rate are necessary and sufficient to rewrite uniquely the n-step Lefkovitch matrix model in terms of the linear combination of the moments. We finally discuss the relationship between the species characteristics of census data and the appropriate types of the Lefkovitch matrix.     

New approximations to the Malthusian parameter

Approximations to the Malthusian parameter of an age-dependent branching process are obtained in terms of the moments of the lifetime distribution, by exploiting a link with renewal theory. In several examples, the new approximations are more accurate than those currently in use, even when based on only the first two moments. The new approximations are extended to include a form of asymmetric cell division that occurs in some species of yeast. When used for inference, the new approximations are shown to have high efficiency.     

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

Continuous-time birth-death Markov processes serve as useful models in population biology. When the birth-death rates are nonlinear, the time evolution of the first n order moments of the population is not closed, in the sense that it depends on moments of order higher than n. For analysis purposes, the time evolution of the first n order moments is often made to be closed by approximating these higher order moments as a nonlinear function of moments up to order n, which we refer to as the moment closure function. In this paper, a systematic procedure for constructing moment closure functions of arbitrary order is presented for the stochastic logistic model. We obtain the moment closure function by first assuming a certain separable form for it, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. The separable structure ensures that the steady-state solutions for the approximate equations are unique, real and positive, while the derivative matching guarantees a good approximation, at least locally in time. Explicit formulas to construct these moment closure functions for arbitrary order of truncation n are provided with higher values of n leading to better approximations of the actual moment dynamics. A host of other moment closure functions previously proposed in the literature are also investigated. Among these we show that only the ones that achieve derivative matching provide a close approximation to the exact solution. Moreover, we improve the accuracy of several previously proposed moment closure functions by forcing derivative matching.     

Stochastic models of kleptoparasitism

In this paper, we consider a model of kleptoparasitism amongst a small group of individuals, where the state of the population is described by the distribution of its individuals over three specific types of behaviour (handling, searching for or fighting over, food). The model used is based upon earlier work which considered an equivalent deterministic model relating to large, effectively infinite, populations. We find explicit equations for the probability of the population being in each state. For any reasonably sized population, the number of possible states, and hence the number of equations, is large. These equations are used to find a set of equations for the means, variances, covariances and higher moments for the number of individuals performing each type of behaviour. Given the fixed population size, there are five moments of order one or two (two means, two variances and a covariance). A normal approximation is used to find a set of equations for these five principal moments. The results of our model are then analysed numerically, with the exact solutions, the normal approximation and the deterministic infinite population model compared. It is found that the original deterministic models approximate the stochastic model well in most situations, but that the normal approximations are better, proving to be good approximations to the exact distribution, which can greatly reduce computing time.     

ON FITTING EQUATIONS TO SENSORY DATA: A POINT OF VIEW, AND A PARADOX IN MODELING AND OPTIMIZING

When fitting equations to data relating ingredients or factor scores to subjective ratings, there are at least two methods to create the equations. One method, (linear) forces in linear terms and allows additional quadratic and interaction terms. The other method, (quadratic) forces in linear and quadratic terms, and then permits cross-terms to enter. The two methods produce contradictory results. The first (and expected) contradiction is that the quadratic model shows optimum levels in the middle range of the levels tested for some, but not all, ingredients. The second (and unexpected) contradiction is that the linear method (which usually does not incorporate many additional terms) generates better validation predictions for " hold-out" samples than does the quadratic method. The differences between the optimum generated by the linear model and the optimum generated by the quadratic model can be quite substantial in terms of expected liking, sensory profile, and image profile.     

New Moment Closures Based on A Priori Distributions with Applications to Epidemic Dynamics

Recently, research that focuses on the rigorous understanding of the relation between simulation and/or exact models on graphs and approximate counterparts has gained lots of momentum. This includes revisiting the performance of classic pairwise models with closures at the level of pairs and/or triples as well as effective-degree-type models and those based on the probability generating function formalism. In this paper, for a fully connected graph and the simple SIS (susceptible-infected-susceptible) epidemic model, a novel closure is introduced. This is done via using the equations for the moments of the distribution describing the number of infecteds at all times combined with the empirical observations that this is well described/approximated by a binomial distribution with time dependent parameters. This assumption allows us to express higher order moments in terms of lower order ones and this leads to a new closure. The significant feature of the new closure is that the difference of the exact system, given by the Kolmogorov equations, from the solution of the newly defined approximate system is of order 1/N(2). This is in contrast with the O(1/N) difference corresponding to the approximate system obtained via the classic triple closure. The fully connected nature of the graph also allows us to interpret pairwise equations in terms of the moments and thus treat closures and the two approximate models within the same framework. Finally, the applicability and limitations of the new methodology is discussed in detail.     

Information loss with transform methods in system identification: a new set of transforms with high information content

Linear transform methods like moments, modulating functions, and Laplace transforms are widely used for parameter estimation in system identification problems because they can reduce a large set of overdetermined equations to a small set of linear and nonlinear equations, which often have a very simple form and a unique solution. However, the effects of noise in the data are neglected in deriving these equations. We show (in terms of Fisher's information measure, the generalized variance, and simulations) that these methods can lead to very large errors in the estimates. We develop a new set of transforms based on the idea of maximizing their Fisher information content. The robustness of these new transforms, in contrast to the others, is illustrated by simulations of nanosecond flourescence decay and multicomponent exponential decay.     

Section Thickness and Grain Count Variation in Tritium Autoradiography

H² styrene-glycolmethacrylate is a suitable standard source for autoradiographic model studies. When it is used for quantitative autoradiography, the coefficient of variation for grain counts is minimal at a section thickness of 2.5 μ and greater. Variation increases with progressively thinner sections. When sections of the same thickness are mounted on separate slides, there is no significant variation in grain counts at the 95% confidence level when the slides are processed simultaneously.     

The number of alleles in multigene families

The probability distribution and moments of the number of alleles present in a sample of homologous chromosomes are studied. It is assumed that there are multiple copies of the gene on each chromosome. When there are only two copies per chromosome or when there are only two or three chromosomes, it is possible to use analytic methods to tackle the problem. Otherwise, a simulation method is suggested.     

Modeling and analysis of stochastic invasion processes

In this paper we derive spatially explicit equations to describe a stochastic invasion process. Parents are assumed to produce a random number of offspring which then disperse according to a spatial redistribution kernel. Equations for population moments, such as expected density and covariance averaged over an ensemble of identical stochastic processes, take the form of deterministic integro-difference equations. These equations describe the spatial spread of population moments as the invasion progresses. We use the second order moments to analyse two basic properties of the invasion. The first property is permanence of form in the correlation structure of the wave. Analysis of the asymptotic form of the invasion wave shows that either (i) the covariance in the leading edge of the wave of invasion asymptotically achieves a permanence of form with a characteristic structure described by an unchanging spatial correlation function, or (ii) the leading edge of the wave has no asymptotic permanence of form with the length scales of spatial correlations continually increasing over time. Which of these two outcomes pertains is governed by a single statistic, φ which depends upon the shape of the dispersal kernel and the net reproductive number. The second property of the invasion is its patchy structure. Patchiness, defined in terms of spatial correlations on separate short (within patch) and long (between patch) spatial scales, is linked to the dispersal kernel. Analysis shows how a leptokurtic dispersal kernel gives rise to patchiness in spread of a population. Received: 11 August 1997 / Revised version: 22 September 1998 / Published online: 4 October 2000     

Probabilistic settling in the Local Exchange Model of turbulent particle transport

The Local Exchange Model (LEM) is a stochastic diffusion model of particle transport in turbulent flowing water. It was developed mainly for application to particles of near-neutral buoyancy that are strongly influenced by turbulent eddies. Turbulence can rapidly transfer such particles to the bed, where settlement can then occur by, for example, sticking to biofilms (e.g., fine particulate organic matter, or FPOM) or attaching to the substrate behaviorally (e.g., benthic invertebrates). Previous papers on the LEM have addressed the problems of how long (time) and far (distance) a suspended particle will be transported before hitting the bed for the first time. These are the hitting-time and hitting-distance problems, respectively. Hitting distances predicted by the LEM for FPOM in natural streams tend to be much shorter than the distances at which most particles actually settle, suggesting that particles usually do not settle the first time they hit the bed. The present paper extends the LEM so it can address probabilistic settling, where a particle encountering the bed can either remain there for a positive length of time (i.e., settle) or immediately reflect back into the water column, each with positive probability. Previous results for the LEM are generalized by deducing a single set of equations governing the probability distribution and moments of a broad class of quantities that accumulate during particle trajectories terminated by hitting or settling on the bed (e.g., transport time, transport distance, cumulative energy expenditure during transport). Key properties of the settling-time and settling-distance distributions are studied numerically and compared with the observed FPOM settling-distance distribution for a natural stream. Some remaining limitations of the LEM and possible means of overcoming them are discussed.     

Oscillations of plants' stems and their damping: theory and experimentation

Free oscillations of upright plants' stems, or in technical terms slender tapered rods with one end free, can be modelled by considering the equilibrium between bending moments and moments resulting from inertia. For stems with apical loads and negligible mass of the stem and for stems with finite mass but without top loading, analytical solutions of the differential equations with appropriate boundary conditions are available for a finite number of cases. For other cases approximations leading to an upper and a lower estimate of the frequency of oscillation omega can be derived. For the limiting case of omega = 0, the differential equations are identical with Greenhill's equations for the stability against Euler buckling of slender poles. To illustrate, the oscillation frequencies of 25 spruce trees (Picea sitchensis (Bong.) Carr.) were compared with those calculated on the basis of their morphology, their density and their static elasticity modulus. For Arundo donax L. and Cyperus alternifolius L. the observed oscillation frequency was used in turn to calculate the dynamic elasticity modulus, which was compared with that determined in three-point bending. Oscillation damping was observed for A. donax and C. alternifolius for plants' stems with and without leaves or inflorescence. In C. alternifolius the difference can be attributed to the aerodynamic resistance of the leaves, whereas in A. donax structural damping in addition plays a major role.     

Simulation of javelin flight using experimental aerodynamic data

This paper discusses computer simulation of the differential equations which describe javelin dynamics in flight. It is shown that the use of experimental aerodynamic forces and moments in the equations is preferable to theoretical approximations for these forces and moments which have been used in previous studies. An example which is characteristic of a good throw is presented and analyzed and many interesting features of the trajectory are pointed out.     

Photon counting statistics analysis of biophotons from hands

The photon counting statistics of biophotons emitted from hands is studied with a view to test its agreement with the Poisson distribution. The moments of observed probability up to seventh order have been evaluated. The moments of biophoton emission from hands are in good agreement while those of dark counts of photomultiplier tube show large deviations from the theoretical values of Poisson distribution. The present results are consistent with the conventional delta-value analysis of the second moment of probability.     

Systematization of a set of closure techniques

Approximations in population dynamics are gaining popularity since stochastic models in large populations are time consuming even on a computer. Stochastic modeling causes an infinite set of ordinary differential equations for the moments. Closure models are useful since they recast this infinite set into a finite set of ordinary differential equations. This paper systematizes a set of closure approximations. We develop a system, which we call a power p closure of n moments, where 0≤p≤n. [Keeling, 2000a] and [Keeling, 2000b] approximation with third order moments is shown to be an instantiation of this system which we call a power 3 closure of 3 moments. We present an epidemiological example and evaluate the system for third and fourth moments compared with Monte Carlo simulations.     

Predicting Physical Activity Energy Expenditure Using Accelerometry in Adults From Sub‐Sahara Africa

Lack of physical activity may be an important etiological factor in the current epidemiological transition characterized by increasing prevalence of obesity and chronic diseases in sub‐Sahara Africa. However, there is a dearth of data on objectively measured physical activity energy expenditure (PAEE) in this region. We sought to develop regression equations using body composition and accelerometer counts to predict PAEE. We conducted a cross‐sectional study of 33 adult volunteers from an urban (n = 16) and a rural (n = 17) residential site in Cameroon. Energy expenditure was measured by doubly labeled water (DLW) over a period of seven consecutive days. Simultaneously, a hip‐mounted Actigraph accelerometer recorded body movement. PAEE prediction equations were derived using accelerometer counts, age, sex, and body composition variables, and cross‐validated by the jack‐knife method. The Bland and Altman limits of agreement (LOAs) approach was used to assess agreement. Our results show that PAEE (kJ/kg/day) was significantly and positively correlated with activity counts from the accelerometer (r = 0.37, P = 0.03). The derived equations explained 14–40% of the variance in PAEE. Age, sex, and accelerometer counts together explained 34% of the variance in PAEE, with accelerometer counts alone explaining 14%. The LOAs between DLW and the derived equations were wide, with predicted PAEE being up to 60 kJ/kg/day below or above the measured value. In summary, the derived equations performed better than existing published equations in predicting PAEE from accelerometer counts in this population. Accelerometry could be used to predict PAEE in this population and, therefore, has important applications for monitoring population levels of total physical activity patterns.     

Specific problems using electronic particle counters

Some specific problems of electronic particle counters (Coulter Counter, Elzone) are discussed. Conductivity of the medium does not influence the size response of the instrument, but might change the size of the particles through osmotic stress. An important problem of electronic particle counters is count loss by coincidence of passage of particles through the orifice and caused by dead time of the instruments. Several correction equations were tried out. It turned out that the count losses were much higher than the ones predicted by the manufacturers. In instruments equipped with multi-channel size analysers dead time count loss is much more important than the count loss by coincidence. A maximal counting frequency of 200 counts/second is recommended for accurate work. The width of the size distribution of particles depended on the diameter of the aperture tube.