On the Rate of Growth of the Population of the United States since 1790 and Its Mathematical Representation

Biological control and the theories of the interactions of populations

Many great men have said that no science is really worthy of the name until it has attained quantitative accuracy and can express its principles and laws in mathematical form.

Progress along developmental tracks for electronic health records implementation in the United States

The development and implementation of electronic health records (EHR) have occurred slowly in the United States. To date, these approaches have, for the most part, followed four developmental tracks: (a) Enhancement of immunization registries and linkage with other health records to produce Child Health Profiles (CHP), (b) Regional Health Information Organization (RHIO) demonstration projects to link together patient medical records, (c) Insurance company projects linked to ICD-9 codes and patient records for cost-benefit assessments, and (d) Consortia of EHR developers collaborating to model systems requirements and standards for data linkage. Until recently, these separate efforts have been conducted in the very silos that they had intended to eliminate, and there is still considerable debate concerning health professionals access to as well as commitment to using EHR if these systems are provided. This paper will describe these four developmental tracks, patient rights and the legal environment for EHR, international comparisons, and future projections for EHR expansion across health networks in the United States.

The interplay between determinism and stochasticity in childhood diseases

An important issue in the history of ecology has been the study of the relative importance of deterministic forces and processes noise in shaping the dynamics of ecological populations. We address this question by exploring the temporal dynamics of two childhood infections, measles and whooping cough, in England and Wales. We demonstrate that epidemics of whooping cough are strongly influenced by stochasticity; fully deterministic approaches cannot achieve even a qualitative fit to the observed data. In contrast, measles dynamics are extremely well explained by a deterministic model. These differences are shown to be caused by their contrasting responses to dynamical noise due to different infectious periods.

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 theta, and thus, dN/dt not only depends on N, but also on theta. So theta 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 theta) 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)=alpha(e)[-0.5(x(i)-x(0)/b)(2)]) 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.

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.

Intraspecific competition and density dependence of food consumption and growth in Arctic charr

1. Intraspecific competition for restricted food resources is considered to play a fundamental part in density dependence of somatic growth and other population characteristics, but studies simultaneously addressing the interrelationships between population density, food acquisition and somatic growth have been missing. 2. We explored the food consumption and individual growth rates of Arctic charr Salvelinus alpinus in a long-term survey following a large-scale density manipulation experiment in a subarctic lake. 3. Prior to the initiation of the experiment, the population density was high and the somatic growth rates low, revealing a severely overcrowded and stunted fish population. 4. During the 6-year period of stock depletion the population density of Arctic charr was reduced with about 75%, resulting in an almost twofold increase in food consumption rates and enhanced individual growth rates of the fish. 5. Over the decade following the density manipulation experiment, the population density gradually rose to intermediate levels, accompanied by corresponding reductions in food consumption and somatic growth rates. 6. The study revealed negative relationships with population density for both food consumption and individual growth rates, reflecting a strong positive correlation between quantitative food intake and somatic growth rates. 7. Both the growth and consumption rate relationships with population density were well described by negative power curves, suggesting that large density perturbations are necessary to induce improved feeding conditions and growth rates in stunted fish populations. 8. The findings demonstrate that quantitative food consumption represents the connective link between population density and individual growth rates, apparently being highly influenced by intraspecific competition for limited resources.

Self-organization of anastral spindles by synergy of dynamic instability, autocatalytic microtubule production, and a spatial signaling gradient

Assembly of the mitotic spindle is a classic example of macromolecular self-organization. During spindle assembly, microtubules (MTs) accumulate around chromatin. In centrosomal spindles, centrosomes at the spindle poles are the dominating source of MT production. However, many systems assemble anastral spindles, i.e., spindles without centrosomes at the poles. How anastral spindles produce and maintain a high concentration of MTs in the absence of centrosome-catalyzed MT production is unknown. With a combined biochemistry-computer simulation approach, we show that the concerted activity of three components can efficiently concentrate microtubules (MTs) at chromatin: (1) an external stimulus in form of a RanGTP gradient centered on chromatin, (2) a feed-back loop where MTs induce production of new MTs, and (3) continuous re-organization of MT structures by dynamic instability. The mechanism proposed here can generate and maintain a dissipative MT super-structure within a RanGTP gradient.

Mors tua, vita mea? The Rise and Fall of Innovative Industrial Clusters

This chapter contains a theoretical analysis of the rise and fall of clusters. It is shown how a major technological innovation sets a process of creative destruction into motion, where new clusters appear and replace clusters based on obsolete technologies. In the last stage when the cluster matures, it either achieves a national or international leadership in a given sector or technology. The decisive element appears to be a different institutional framework. Using simulation techniques, it is shown how policies that support firm-based micro-level incentives seem to be critical rather than policies aimed at strengthening the ‘carrying capacities’. Most European policy makers overemphasize the latter type of policies as a means to initiate cluster emergence and growth.

An alternative formulation for a delayed logistic equation

We derive an alternative expression for a delayed logistic equation, assuming that the rate of change of the population depends on three components: growth, death, and intraspecific competition, with the delay in the growth component. In our formulation, we incorporate the delay in the growth term in a manner consistent with the rate of instantaneous decline in the population given by the model. We provide a complete global analysis, showing that, unlike the dynamics of the classical logistic delay differential equation (DDE) model, no sustained oscillations are possible. Just as for the classical logistic ordinary differential equation (ODE) growth model, all solutions approach a globally asymptotically stable equilibrium. However, unlike both the logistic ODE and DDE growth models, the value of this equilibrium depends on all of the parameters, including the delay, and there is a threshold that determines whether the population survives or dies out. In particular, if the delay is too long, the population dies out. When the population survives, i.e., the attracting equilibrium has a positive value, we explore how this value depends on the parameters. When this value is positive, solutions of our DDE model seem to be well approximated by solutions of the logistic ODE growth model with this carrying capacity and an appropriate choice for the intrinsic growth rate that is independent of the initial conditions.

Comportement contre-intuitif d'un modèle mécaniste de succession végétale

A model of plant succession has been built for simulating the growth of six species, based on physiological processes such as photosynthesis, growth and maintenance respiration, carbon allocation, nitrogen absorption, fixation and remobilisation, organ mortality, seed dispersal and germination. The simulated global plant productivity decreases in response to an increase in soil nitrogen availability. The reason is a replacement of the rapidly-growing legume species by slow-growing shrub species that are higher than the legume and thus shade it. This paradoxical result could have a wide application field.

On moment closures for population dynamics in continuous space

A first-order moment closure, the mean-field assumption that organisms encounter one another in proportion to their spatial average densities, lies at the heart of much theoretical ecology. This assumption ignores all spatial information and, at the very least, needs to be replaced by a second-order closure to gain understanding of ecological dynamics in spatially structured populations. We describe a number of conditions that a second-order closure should satisfy and use these conditions to evaluate some closures currently available in the literature. Two conditions are particularly helpful in discriminating among the alternatives: that the closure should be positive, and that the dynamics should be unaltered when identical individuals are given different labels. On this basis, a class of closures we refer to as 'power-2' turns out to provide a good compromise between positivity and dynamical invariance under relabelling.

Predicting polymorphic transformation curves using a logistic equation

The commonly used solid-state reaction models (for example-Prout-Tompkins, Avrami-Erofe'ev) describe the polymorphic transformation data only over a certain range, alpha from 10% to 90%. Predictions based on a fit to a fraction of the data are inadequate because we ignore the early induction phase of the reaction, which is important for predictive purposes. A four-parameter logistic equation describes the data over the entire curve for polymorphic transformation at high temperatures. We use the parameters of the logistic equation to predict the transformation curves. The predicted curves agree with the experimental data.

Plant Defense and Density Dependence in the Population Growth of Herbivores

Long-standing theory has predicted that plant defensive and nutritional traits contribute to the population dynamics of insect herbivores. To examine the role of plant variation in density dependence, I took a comparative approach by conducting density manipulation experiments with the specialist aphid, Aphis nerii, on 18 species of milkweed (Asclepias spp.). The strength of density dependence varied on the plant species. Variation in plant secondary compounds (cardenolides), trichomes, leaf carbon and nitrogen concentrations, and seed mass of the milkweed species predicted the R(max) of aphid populations, while specific leaf weight, carbon concentration, latex, water content, and trichome density were significant predictors of the strength of density dependence. Thus, plant traits that probably evolved for primary and defensive functions contribute to the ecological dynamics of herbivore populations.

Threat of predation negates density effects in larval gray treefrogs

While density-dependence is central to most theory regarding population regulation and community structure, specific mechanisms that modify its effects in the absence of changes in consumer-resources ratios (e.g., thinning) are not well understood. To determine if the threat of predation alters effects of density, we investigated the interaction between density of larval treefrogs (Hyla chrysoscelis) and the non-lethal presence of a predatory fish (Enneacanthus obesus). A significant density by fish interaction was consistent for all response variables (e.g., larval survivorship, mass, and time to metamorphosis) driven by a complete lack of density effects in the presence of predators, while predator-free tanks showed classic density-dependent responses. Given that female H. chrysoscelis strongly avoid ovipositing in ponds containing fish, certain larval adaptations are apparently not constrained by maternal behavior and suggest redundancy in response to predators. Our data suggest that non-lethal effects of predators can determine larval performance irrespective of larval density, and that the non-lethal effects of predators can be strong whether lethal effects are strong or weak.

A functionally diverse population growth model

The present work deals with a single species population growth model, exhibiting diverse functional modes. The model is based on the principle of limiting factors for population growth. This paradigm was adapted from the law of the minimum, and the law of the tolerance. In the framework presented here, rates of natality and mortality are controlled by extreme values of limiting factors. The formal expression of these ideas corresponds to what we call a functionally diverse model. This model permits the identification of thresholds of viability, starvation, intraspecific cooperation and competition. The performance of the model is evaluated using data for populations growing under experimental or natural conditions.

Effect of environmental fluctuations on the dynamic composition of engineered cartilage: A deterministic model in stochastic environment

Dynamics of extracellular matrix (ECM) deposition and scaffold degradation in cell-polymer constructs have been studied in a random fluctuating environment created due to the applications of growth factors into the in vitro generation of cartilaginous constructs. Existing models of cell-polymer constructs for the design of engineered cartilage have been discussed and then a new deterministic scheme in random environment proposed taking into account the effects of growth factors as the environmental variability in the form of Gaussian white noise. Steady-state probability distribution of each individual component of the ECM in its homeostasis is found explicitly. The computer-simulated results of the model have been discussed and then compared with the data from a variety of scaffold systems and culture conditions.

Analysis of the logistic function model: derivation and applications specific to batch cultured microorganisms

Mathematical models are useful for describing microbial growth, both in natural ecosystems and under research conditions. To this end, a rate expression that accounted for depletion of nutrients was used to derive the logistic function model for batch cultures. Statistical analysis was used to demonstrate the suitability of this model for growth curve data. Two linear forms of the model and two procedures for calculating growth rate constants were derived to facilitate statistical evaluation of growth curves. The procedures for calculating growth rate constants were found to be useful for calculation of growth rate constants at each time point, or for estimating growth rate constants from early growth curve data. The utility of the logistic function model and its alternative forms is discussed with respect to planning experiments, analyzing growth curves for the effects of factors other than nutrient limitation, and developing more complete descriptions of cell proliferation.

Population growth rate and its determinants: an overview

We argue that population growth rate is the key unifying variable linking the various facets of population ecology. The importance of population growth rate lies partly in its central role in forecasting future population trends; indeed if the form of density dependence were constant and known, then the future population dynamics could to some degree be predicted. We argue that population growth rate is also central to our understanding of environmental stress: environmental stressors should be defined as factors which when first applied to a population reduce population growth rate. The joint action of such stressors determines an organism's ecological niche, which should be defined as the set of environmental conditions where population growth rate is greater than zero (where population growth rate = r = log(e)(N(t+1)/N(t))). While environmental stressors have negative effects on population growth rate, the same is true of population density, the case of negative linear effects corresponding to the well-known logistic equation. Following Sinclair, we recognize population regulation as occurring when population growth rate is negatively density dependent. Surprisingly, given its fundamental importance in population ecology, only 25 studies were discovered in the literature in which population growth rate has been plotted against population density. In 12 of these the effects of density were linear; in all but two of the remainder the relationship was concave viewed from above. Alternative approaches to establishing the determinants of population growth rate are reviewed, paying special attention to the demographic and mechanistic approaches. The effects of population density on population growth rate may act through their effects on food availability and associated effects on somatic growth, fecundity and survival, according to a 'numerical response', the evidence for which is briefly reviewed. Alternatively, there may be effects on population growth rate of population density in addition to those that arise through the partitioning of food between competitors; this is 'interference competition'. The distinction is illustrated using a replicated laboratory experiment on a marine copepod, Tisbe battagliae. Application of these approaches in conservation biology, ecotoxicology and human demography is briefly considered. We conclude that population regulation, density dependence, resource and interference competition, the effects of environmental stress and the form of the ecological niche, are all best defined and analysed in terms of population growth rate.

Analysis of logistic growth models

A variety of growth curves have been developed to model both unpredated, intraspecific population dynamics and more general biological growth. Most predictive models are shown to be based on variations of the classical Verhulst logistic growth equation. We review and compare several such models and analyse properties of interest for these. We also identify and detail several associated limitations and restrictions.A generalized form of the logistic growth curve is introduced which incorporates these models as special cases. Several properties of the generalized growth are also presented. We furthermore prove that the new growth form incorporates additional growth models which are markedly different from the logistic growth and its variants, at least in their mathematical representation. Finally, we give a brief outline of how the new curve could be used for curve-fitting.

Growth curve analyses in selected duck lines

1. Growth patterns of male ducks from 4 lines (lines A, B, C and D) selected for market weight were analysed and compared to growth patterns of ducks in the respective line 7 generations earlier. Growth curves were analysed using procedures derived from the Weibull sigmoidal function and the linear-linear relative growth rate model and simple allometry. 2. The ducks were fed ad libitum under 24-h lighting throughout the experiment. At weekly intervals from the time of hatch through 70 d of age, 16 ducks from each line were killed to determine body, carcase, breast-muscle, leg and thigh-muscle, and abdominal fat weights. 3. Line A was the heaviest line, followed by line B, line C and line D. However, body weight, carcase weight and breast-muscle weight at 49 d of age were not significantly different between lines A and B. After 7 generations of selection, the breast-muscle yield was increased to >19% and the abdominal fat percent was reduced to <1.4% in all lines. 4. The Weibull growth curve analysis of body weight showed an increase in the asymptotes during selection, while the age of the inflection point remained constant in all lines (21.3 to 26.0 d). For breast-muscle growth, ducks reached the inflection point 12.8 to 14.3 d later than for body weight. Between line A and line B, asymptotes for body weight, asymptotes for breast-muscle weight and allometric growth coefficients of breast muscle and leg and thigh muscles from 14 to 49 d were not significantly different. 5. The relative growth rate model discriminated body and breast-muscle growth patterns of line A and line B. The initial decline in the relative body growth rate was less and the time to reach the transition was longer in line A than line B. On the other hand, the initial decline in the relative breast-muscle growth rate was greater in line A than line B.

Extinction and quasi-stationarity in the Verhulst logistic model

We formulate and analyse a stochastic version of the Verhulst deterministic model for density-dependent growth of a single population. Three parameter regions with qualitatively different behaviours are identified. Explicit approximations of the quasi-stationary distribution and of the expected time to extinction are presented in each of these regions. The quasi-stationary distribution is approximately normal, and the time to extinction is long, in one of these regions. Another region has a short time to extinction and a quasi-stationary distribution that is approximately truncated geometric. A third region is a transition region between these two. Here the time to extinction is moderately long and the quasi-stationary distribution has a more complicated behaviour. Numerical illustrations are given.

Insect biodemography

Biodemography is an emerging subdiscipline of classical demography that brings life table techniques, mortality models, experimental systems, and comparative methods to bear on questions concerned with the fundamental determinants of mortality, longevity, aging, and life span. It is important to entomology because it provides a secure and comprehensive actuarial foundation for life table and mortality analysis, it suggests new possibilities for the use of model insect systems in the study of aging and mortality dynamics, and it integrates an interdisciplinary perspective on demographic concepts and actuarial techniques into the entomological literature. This paper describes the major life table formulae and mortality models used to analyze the actuarial properties of insects; summarizes the literature on adult insect life span, including a discussion of basic concepts; identifies the major correlates of extended longevity; and suggests new ideas for using demographic concepts in both basic and applied entomology.

On competition between different species of graminivorous insects

The growth of pure populations of the beetles Rhizopertha dominica and Oryzaephilus surinamensis , and of the moth Sitotroga cerealella , was observed in a standard medium of wheat. This was maintained at a constant level by the periodic removal of ‘conditioned’ frass and the addition of fresh grains. The population of each species rose to a maximum and remained fluctuating about this value indefinitely. A comparison of the rates of oviposition, with the rates at which adults emerged, showed that in the maximum population there was an enormous mortality (always over 90%) in the immature stages. When pairs of species competed Rhizopertha eliminated Sitotroga because their larvae, between which most of the competition occurred, have the same needs and habits. But each of these species was able to survive with Oryzaephilus because this species occupies a different ‘ecological niche’. The Verhulst-Pearl ‘logistic’ equation (1), for the growth of population of a single species in a limited environment, and the Lotka-Volterra simultaneous equations (2), for the growth of population of two species competing for the same limited environment, were fitted to the census data from all the experiments. The biological assumptions on which they are based proved to be true for practical purposes for Rhizopertha and Sitotroga populations. These assumptions are that the value of the potential rate of increase remains statistically constant and that all the factors inhibiting increase are linearly related to population density. Further­ more, a single factor, larval competition, was represented by the single indices standing for interspecific inhibition. It follows that the maximum population (or equilibrium position) should be independent of the initial population, and this proved to be so for all species. Equations (2) did not always fit the observed points very well, but they were always success­ful in predicting the outcome of competition. It does not follow from this that these equa­tions have any general validity. Their basic assumptions are by no means universally true and, unless they are shown to be so for a particular species under known, environmental conditions, no biological deductions can be drawn from them. Where they do apply they describe the course of change of population of two competing organisms with an accuracy which depends on the constancy of the coefficients involved. Two kinds of organism will be able to survive together only if they differ in needs and habits, i. e. occupy different ecological niches. Populations living in a medium of unrenewed wheat rose to a maximum and then declined as the food became exhausted and ‘conditioning’ increased. The eventual extinction of the population was due, not to the cessation of oviposition, but to the failure of the larvae to survive. The longevity of Rhizopertha adults was lower in unrenewed than in renewed medium, and lower still when this species was competing with Sitotroga in unrenewed medium. The longevity of the other species, and the sex ratio of Sitotroga , were apparently unaffected by these conditions. The fecundity of Rhizopertha females decreased with time, and the length of Sitotroga adults of both sexes decreased in succeeding generations. The competitive relation­ship between both Sitotroga and Rhizopertha , and Oryzaephilus shifted slightly in favour of the former species in unrenewed as compared with renewed media. In a renewed medium this relationship probably depends chiefly on the destruction of eggs and pupae by adults and larvae, for which the more predaceous Oryzaephilus is better placed. In an unrenewed medium the ability of the larvae to make the best use of the limited food supply is the determining factor, and here the other two species have the advantage. The competitive relationship between Rhizopertha and Sitotroga remained the same in both media.