The application of mass and energy conservation laws in physiologically structured population models of heterotrophic organisms

Rules for energy uptake, and subsequent utilization, form the basis of population dynamics and, therefore, explain the dynamics of the ecosystem structure in terms of changes in standing crops and size distributions of individuals. Mass fluxes are concomitant with energy flows and delineate functional aspects of ecosystems by defining the roles of individuals and populations. The assumption of homeostasis of body components, and an assumption about the general structure of energy budgets, imply that mass fluxes can be written as weighted sums of three organizing energy fluxes with the weight coefficients determined by the conservation law of mass. These energy fluxes are assimilation, maintenance and growth, and provide a theoretical underpinning of the widely applied empirical method of indirect calorimetry, which relates dissipating heat linearly to three mass fluxes: carbon dioxide production, oxygen consumption and N-waste production. A generic approach to the stoichiometry of population energetics from the perspective of the individual organism is proposed and illustrated for heterotrophic organisms. This approach indicates that mass transformations can be identified by accounting for maintenance requirements and overhead costs for the various metabolic processes at the population level. The theoretical background for coupling the dynamics of the structure of communities to nutrient cycles, including the water balance, as well as explicit expressions for the dissipating heat at the population level are obtained based on the conservation law of energy. Specifications of the general theory employ the Dynamic Energy Budget model for individuals. Copyright 1999 Academic Press.     

Population variation and individual maximum size in two leech populations: energy extraction from cannibalism or niche widening?

The theory of cannibal dynamics predicts a link between population dynamics and individual life history. In particular, increased individual growth has, in both modeling and empirical studies, been shown to result from a destabilization of population dynamics. We used data from a long-term study of the dynamics of two leech (Erpobdella octoculata) populations to test the hypothesis that maximum size should be higher in a cycling population; one of the study populations exhibited a delayed feedback cycle while the other population showed no sign of cyclicity. A hump-shaped relationship between individual mass of 1-year-old leeches and offspring density the previous year was present in both populations. As predicted from the theory, the maximum mass of individuals was much larger in the fluctuating population. In contrast to predictions, the higher growth rate was not related to energy extraction from cannibalism. Instead, the higher individual mass is suggested to be due to increased availability of resources due to a niche widening with increased individual body mass. The larger individual mass in the fluctuating population was related to a stronger correlation between the densities of 1-year-old individuals and 2-year-old individuals the following year in this population. Although cannibalism was the major mechanism regulating population dynamics, its importance was negligible in terms of providing cannibalizing individuals with energy subsequently increasing their fecundity. Instead, the study identifies a need for theoretical and empirical studies on the largely unstudied interplay between ontogenetic niche shifts and cannibalistic population dynamics.     

Body weight setpoint, metabolic adaption and human starvation

A biological setpoint for fatness has been proposed in the medical literature. This body weight setpoint functions as a point of stable equilibrium. In an underfed state, with resulting weight loss, the body will reduce the relative energy expenditure by metabolic adaption which reduces the rate of weight loss. Previous mathematical models of energy expenditure and weight loss dynamics have not addressed this setpoint mechanism. The setpoint model has been proposed to quantify this biological process and is unique in predicting energy expenditure during weight loss as a function of the setpoint fat-free mass ratio and setpoint energy expenditure, eliminating the various controlling characteristics such as age, gender and heredity. The model is applied to the seminal Minnesota human semistarvation experiment and is used to predict weight vs time on an individual basis and the caloric requirements for weight maintenance at the reduced weight. Comparison is made with the Harris-Benedict equations and the Brody-Kleiber. (W ^3/4) law.     

Dynamic energy budget theory and population ecology: lessons from Daphnia

Dynamic energy budget (DEB) theory offers a perspective on population ecology whose starting point is energy utilization by, and homeostasis within, individual organisms. It is natural to ask what it adds to the existing large body of individual-based ecological theory. We approach this question pragmatically--through detailed study of the individual physiology and population dynamics of the zooplankter Daphnia and its algal food. Standard DEB theory uses several state variables to characterize the state of an individual organism, thereby making the transition to population dynamics technically challenging, while ecologists demand maximally simple models that can be used in multi-scale modelling. We demonstrate that simpler representations of individual bioenergetics with a single state variable (size), and two life stages (juveniles and adults), contain sufficient detail on mass and energy budgets to yield good fits to data on growth, maturation and reproduction of individual Daphnia in response to food availability. The same simple representations of bioenergetics describe some features of Daphnia mortality, including enhanced mortality at low food that is not explicitly incorporated in the standard DEB model. Size-structured, population models incorporating this additional mortality component resolve some long-standing questions on stability and population cycles in Daphnia. We conclude that a bioenergetic model serving solely as a 'regression' connecting organismal performance to the history of its environment can rest on simpler representations than those of standard DEB. But there are associated costs with such pragmatism, notably loss of connection to theory describing interspecific variation in physiological rates. The latter is an important issue, as the type of detailed study reported here can only be performed for a handful of species.     

Body mass–abundance relationships are robust to cascading effects in marine food webs

Body mass has been shown to scale negatively with abundance in a wide range of habitats and ecosystems. It is believed that this relationship has important consequences for the distribution and maintenance of energy in natural communities. Some studies have shown that the relationship between body mass and abundance may be robust to major food web perturbations, fuelling the belief that natural processes may preserve the slope of this relationship and the associated cycling of energy and nutrients. Here, we use data from a long‐term experimental food web manipulation to examine this issue in a semi‐natural environment. Similar communities were developed in large experimental mesocosms over a six month period. Some of the mesocosms were then subjected to species removals, based on the mean strength of their trophic interactions in the communities. In treatments where the strongest interactors were removed, a community‐level trophic cascade occurred. The biomass density of invertebrates increased dramatically in these communities, which led to a suppression of primary production. In spite of these widespread changes in ecosystem functioning, the slope of the relationship between body mass and abundance remained unchanged. This was the case whether average species body mass and abundance or individual organism size spectra were considered. An examination of changes in species composition before and after the experimental manipulations revealed an important mechanism for maintaining the body mass–abundance relationship. The manipulated communities all had a higher species turnover than the intact communities, with the highest turnover in communities that experienced cascading effects. As some species increased in body mass and abundance, new species filled the available size–abundance niches that were created. This maintained the overall body mass–abundance relationship and provided a stabilising structure to these experimental communities.     

小型哺乳动物的母体效应及其在种群调节中的作用

母体效应是指双亲的表型影响其后代表型的直接效应。它是子代对环境异质性的一种表型反应,亦是进化动力的一个重要来源,还可能与小型哺乳动物种群调节机制有关。以小型哺乳动物为例,介绍了母体效应的概念及其产生和发展过程,以及影响母体效应的营养和非营养因素,特别强调了光周期和激素的作用。在种群水平上,对度量母体效应的备选指标进行了评价,认为种群内个体的平均体重能较好地代表种群质量的高低;概述了衰老母体效应假说的主要内容及其在小型哺乳动物种群动态调节中的作用,即在种群数量的周期性波动过程中,母体质量的变化会影响后代的生殖和存活,甚至持续达2～3个世代,它与由种群年龄结构偏移所导致的衰老效应共同起作用,可使某些小型哺乳动物种群处于低数量期。本文还对母体效应的进化适应意义进行丁阐述。     

Predicting changes of body weight, body fat, energy expenditure and metabolic fuel selection in C57BL/6 mice

The mouse is an important model organism for investigating the molecular mechanisms of body weight regulation, but a quantitative understanding of mouse energy metabolism remains lacking. Therefore, we created a mathematical model of mouse energy metabolism to predict dynamic changes of body weight, body fat, energy expenditure, and metabolic fuel selection. Based on the principle of energy balance, we constructed ordinary differential equations representing the dynamics of body fat mass (FM) and fat-free mass (FFM) as a function of dietary intake and energy expenditure (EE). The EE model included the cost of tissue deposition, physical activity, diet-induced thermogenesis, and the influence of FM and FFM on metabolic rate. The model was calibrated using previously published data and validated by comparing its predictions to measurements in five groups of male C57/BL6 mice (N = 30) provided ad libitum access to either chow or high fat diets for varying time periods. The mathematical model accurately predicted the observed body weight and FM changes. Physical activity was predicted to decrease immediately upon switching from the chow to the high fat diet and the model coefficients relating EE to FM and FFM agreed with previous independent estimates. Metabolic fuel selection was predicted to depend on a complex interplay between diet composition, the degree of energy imbalance, and body composition. This is the first validated mathematical model of mouse energy metabolism and it provides a quantitative framework for investigating energy balance relationships in mouse models of obesity and diabetes.     

The cost of enzyme synthesis in the genetics of energy balance and physiological performance

The study of metabolism has traditionally focused upon factors that influence metabolic rate, at levels of both the metabolic pathway and the whole organism. This paper focuses on the cost, and thereby the efficiency, of metabolic processes. The genotype-dependent cost of enzyme turnover is proposed as a biochemical genetic mechanism for relating genetic variation at single genes to phenotypic variation in quantitative traits of energy metabolism. Decreased costs of maintenance metabolism can accompany artificial selection for increased production (e.g. growth, reproduction, etc.) and lower maintenance is correlated with multiple locus heterozygosity in outbred populations. In both cases, high production has been associated with lower rates of protein turnover. Several factors influence the ATP-equivalent cost of enzyme turnover. These factors are used to calculate the cost of turnover for a single enzyme. This cost can conservatively constitute up to several percent of the total daily mass-specific energy demands of maintenance metabolism. Genetic variants of an enzyme can differ in the cost of turnover. These differences can constitute the basis for metabolic changes associated with artificial selection for production and the metabolic differences that are associated with individual levels of heterozygosity. The metabolic and evolutionary significance of genotype-dependent turnover costs is a function of individual energy balance. The strength of selection against increases in cost will be an inverse function of individual energy balance and is therefore influenced by both environmental and genetic factors.     

Dynamics of starvation in humans

A differential equation model describing the dynamics of stored energy in the form of fat mass, lean body mass and ketone body mass during prolonged starvation is developed. The parameters of the model are estimated using available data for 7 days into starvation. A simulation of energy stores for a normal individual with body mass index between 19 and 24 and an obese individual with body mass index over 30 are calculated. The length of time the obese subject can survive during prolonged starvation is estimated using the model. Authors are listed in alphabetical order.     

Mechanistic model of the growth of Streptococcus cremoris HP at super optimal temperatures

The kinetic response of Streptococcus cremoris HP to growth at super optimal temperatures is reported. The response to a step increase in temperature was shown to be transient and to result from an increased metabolic rate caused by the raised temperature combined with thermal deactivation of the cell mass present. The catabolic and anabolic activities of the cell were shown to decay at different rates resulting in an accumulation of cells capable of catabolism (energy production) but unable to reproduce. The proposed mechanism was confirmed by independent estimates of the catabolic and anabolic activities of the culture. A mathematical model based on the proposed mechanism and incorporating simultaneous exponential growth, thermal death, and catabolic uncoupling of anabolically inactive cells was developed. Experimental evaluation of the model indicated the presence of a delay in deactivation of metabolic activity in response to a temperature transient. After the inclusion of this delay in death, it was confirmed that the model was capable of prediction of the balanced growth and transient response of this organism to changes in growth temperature. The delay in death was shown to be of major significance to the control of a simulated cheddar cheese fermentation.     

Population cycles caused by selection by density dependent competitive interactions

Several animal species have cyclic population dynamics with phase-related cycles in life history traits such as body mass, reproductive rate, and pre-reproductive period. Although many mechanisms have been proposed there is no agreement on the cause of these cycles, and no population equation that deduces both the abundance and the life history cycles from basic ecological constraints has been formulated. Here I deduce a population dynamic equation from the selection pressure of density dependent competitive interactions in order to explain the cyclic dynamics in abundance and life history traits. The model can explain cycles by evolutionary changes in the genotype or by plastic responses in the phenotype. It treats the population dynamic growth rate as an initial condition, and its density independent fundament is Fisher’s (1930, The Genetical Theory of Natural Selection, Oxford: Clarendon) fundamental theorem of natural selection that predicts a hyper-geometrical increase in abundance. The predicted periods coincide with the cyclic dynamics of Lepidoptera, and the Calder hypothesis, which suggests that the period of population cycles is proportional to the 1/4 power of body mass, follows from first principles of the proposed density dependent ecology.     

Bird diversity and environmental gradients in Britain: a test of the species–energy hypothesis

1.  We tested the species diversity–energy hypothesis using the British bird fauna. This predicts that temperature patterns should match diversity patterns. We also tested the hypothesis that the mechanism operates directly through effects of temperature on thermoregulatory loads; this further predicts that seasonal changes in temperature cause matching changes in patterns of diversity, and that species' body mass is influential.
2.  We defined four assemblages using migration status (residents or visitors) and season (summer or winter distribution). Records of species' presence/absence in a total of 2362, 10 × 10-km, quadrats covering most of Britain were used, together with a wide selection of habitat, topographic and seasonal climatic data.
3.  We fitted a logistic regression model to each species' distribution using the environmental data. We then combined these individual species models mathematically to form a diversity model. Analysis of this composite model revealed that summer temperature was the factor most strongly associated with diversity.
4.  Although the species–energy hypothesis was supported, the direct mechanism, predicting an important role for body mass and matching seasonal patterns of change between diversity and temperature, was not supported.
5.  However, summer temperature is the best overall explanation for bird diversity patterns in Britain. It is a better predictor of winter diversity than winter temperature. Winter diversity is predicted more precisely from environmental factors than summer diversity.
6.  Climate change is likely to influence the diversity of different areas to different extents; for resident species, low diversity areas may respond more strongly as climate change progresses. For winter visitors, higher diversity areas may respond more strongly, while summer visitors are approximately neutral.     

Effects of Growth Curve Plasticity on Size-Structured Population Dynamics

The physiological-structured population models assume that a fixed fraction of energy intake is utilized for individual growth and maintenance while the remaining for adult fertility. The assumption results in two concerns: energy loss for juveniles and a reproduction dilemma for adults. The dilemma results from the possibility that adults have to breed even if metabolic costs fail to be covered. We consider a size-structured population model, where standard metabolism is given top priority for utilizing energy intake and the surplus energy, if there is any, is distributed to individual growth and reproduction. Moreover, the portion of surplus energy for reproduction is size-dependent and increases monotonically with size. Using the newly developed parameter continuation, we demonstrate their disparate effects on population dynamics. Results show that the size-dependent mechanism of energy allocation primarily exerts destabilizing effects on the system but considerably promotes species coexistence, in comparison with the size-independent mechanism. We conclude that the size-dependent mechanism is, to a large extent, a dispensable component of model ingredients when ontogeny is explicitly taken into consideration.     

Daily torpor is associated with telomere length change over winter in Djungarian hamsters

Ageing can progress at different rates according to an individual's physiological state. Natural hypothermia, including torpor and hibernation, is a common adaptation of small mammals to survive intermittent or seasonal declines in environmental conditions. In addition to allowing energy savings, hypothermia and torpor have been associated with retarded ageing and increased longevity. We tested the hypothesis that torpor use slows ageing by measuring changes in the relative telomere length (RTL) of Djungarian hamsters, Phodopus sungorus, a highly seasonal rodent using spontaneous daily torpor, over 180 days of exposure to a short-day photoperiod and warm (approx. 20°C) or cold (approx. 9°C) air temperatures. Multi-model inference showed that change in RTL within individuals was best explained by positive effects of frequency of torpor use, particularly at low body temperatures, as well as the change in body mass and initial RTL. Telomere dynamics have been linked to future survival and proposed as an index of rates of biological ageing. Our results therefore support the hypothesis that daily torpor is associated with physiological changes that increase somatic maintenance and slow the processes of ageing.     

A Dynamic Energy Budget model based on partitioning of net production

We formulate a Dynamic Energy Budget (DEB) model for the growth and reproduction of individual organisms based on partitioning of net production (i.e. energy acquisition rate minus maintenance rate) between growth and energy reserves. Reproduction uses energy from reserves. The model describes both feeding and non-feeding stages, and hence is applicable to embryos (which neither feed nor reproduce), juveniles (which feed but do not reproduce), and adults (which commonly both feed and reproduce). Embryonic growth can have two forms depending on the assumptions for acquisition of energy from yolk. By default, when the energy acquisition rate exceeds the maintenance rate, a fixed proportion of the resulting net production is spent on growth (increase in structural biomass), and the remaining portion is channelled to the reserves. Feeding organisms, however, modulate their allocation of net production energy in response to their total energy content (energy in the reserves plus energy bounded to structural biomass). In variable food environment an organism alternates between periods of growth, no-growth, and balanced-growth. In the latter case the organism adopts an allocation strategy that keeps its total energy constant. Under constant environmental conditions, the growth of a juvenile is always of von Bertalanffy type. Depending on the values of model parameters there are two long-time possibilities for adults: (a) von Bertalanffy growth accompanied by reproduction at a rate that approaches zero as the organism approaches asymptotic size, or (b) abrupt cessation of growth at some finite time, following which, the rate of reproduction is constant. We illustrate the model's applicability in life history theory by studying the optimum values of the energy allocation parameters for constant environment and for each of the dynamic regimes described above. Received: 11 May 1998 / Revised version: 18 February 2000 / Published online: 4 October 2000     

An Individual-Based Model of Zebrafish Population Dynamics Accounting for Energy Dynamics

Developing population dynamics models for zebrafish is crucial in order to extrapolate from toxicity data measured at the organism level to biological levels relevant to support and enhance ecological risk assessment. To achieve this, a dynamic energy budget for individual zebrafish (DEB model) was coupled to an individual based model of zebrafish population dynamics (IBM model). Next, we fitted the DEB model to new experimental data on zebrafish growth and reproduction thus improving existing models. We further analysed the DEB-model and DEB-IBM using a sensitivity analysis. Finally, the predictions of the DEB-IBM were compared to existing observations on natural zebrafish populations and the predicted population dynamics are realistic. While our zebrafish DEB-IBM model can still be improved by acquiring new experimental data on the most uncertain processes (e.g. survival or feeding), it can already serve to predict the impact of compounds at the population level.     

Seasonal and developmental changes of reproductive organs of male ringed seals (Phoca hispida) in the Svalbard area

The weights of testes, prostate gland and baculum of ringed seal males were related to age, season and differences in body size. There was a significant seasonal variation in testes and prostate gland size of sexually mature males, with a maximum occurring in early April. There were no seasonal changes in prostate weight of immature males, but some of the older immatures had elevated testes weights in April. Testes weight was significantly correlated with lean body mass. The increase in testes size with increasing body weight was greater for seals six years of age or older than for younger males. We suggest that some testicular growth and a seasonal cycle in testes growth occur before the testes become functional endocrinologically. We also believe that the primary event leading to puberty in ringed seals is an age-dependent shift in metabolic processes, directing a larger percentage of available energy towards the reproductive organs.     

No evidence of juvenile body mass affecting dispersal in male red deer

Dispersal is an important mechanism in population dynamics with a sparse empirical basis. Environmental causes of dispersal may work directly or indirectly. In a population with documented negative density-dependent male dispersal, we investigated if the effect of density on dispersal was indirectly mediated through body mass. We analysed the probability of dispersal in 170 juvenile red deer males in Snillfjord municipality, Norway, during a 20-year period of rapid population growth (1977–1997). Body mass and dispersal propensity were not related. Thus, changes in population density act directly on dispersal and are not affected by body mass. Body mass-dependent dispersal occurs in species with strong antagonistic interactions and a high cost of dispersal. Our result suggests that the cost of dispersal in male red deer is low in terms of energy expenditure and survival. We conclude that the effect of body mass on dispersal is likely to vary with mating system and cost of dispersal.     

A teleonomic model describing performance (body, milk and intake) during growth and over repeated reproductive cycles throughout the lifespan of dairy cattle. 1. Trajectories of life function priorities and genetic scaling

The prediction of the control of nutrient partitioning, particularly energy, is a major issue in modelling dairy cattle performance. The proportions of energy channelled to physiological functions (growth, maintenance, gestation and lactation) change as the animal ages and reproduces, and according to its genotype and nutritional environment. This is the first of two papers describing a teleonomic model of individual performance during growth and over repeated reproductive cycles throughout the lifespan of dairy cattle. The conceptual framework is based on the coupling of a regulating sub-model providing teleonomic drives to govern the work of an operating sub-model scaled with genetic parameters. The regulating sub-model describes the dynamic partitioning of a mammal female's priority between life functions targeted to growth (G), ageing (A), balance of body reserves (R) and nutrient supply of the unborn (U), newborn (N) and suckling (S) calf. The so-called GARUNS dynamic pattern defines a trajectory of relative priorities, goal directed towards the survival of the individual for the continuation of the specie. The operating sub-model describes changes in body weight (BW) and composition, foetal growth, milk yield and composition and food intake in dairy cows throughout their lifespan, that is, during growth, over successive reproductive cycles and through ageing. This dynamic pattern of performance defines a reference trajectory of a cow under normal husbandry conditions and feed regimen. Genetic parameters are incorporated in the model to scale individual performance and simulate differences within and between breeds. The model was calibrated for dairy cows with literature data. The model was evaluated by comparison with simulations of previously published empirical equations of BW, body condition score, milk yield and composition and feed intake. This evaluation showed that the model adequately simulates these production variables throughout the lifespan, and across a range of dairy cattle genotypes.     

Informational hypothesis of aging: How does the germ line “avoid” aging?

An informational hypothesis of aging is formulated, and a model is proposed for the survival of a population of multicellular organisms under conditions of informational degradation of the cell genetic material (accumulation of random errors in the genome). The germline cells are not anyhow privileged in this respect, but the genetic material undergoes “rejuvenation” (decrease in the number of errors) during its transition from parents to progeny, owing to meiotic crossover (which changes the error density in individual gametes) and subsequent selection of the most functional (i.e. most error-free) gametes to form the progeny genomes. With a simulation-type model, it is shown that such a mechanism ensures stability of the population gene pool in many generations, though in any individual organism the amount of genome errors in all cells increases with aging.