PATTERNS OF GROWTH IN BIRDS. II. GROWTH RATE AND MODE OF DEVELOPMENT

This analysis was initiated to examine the relationship between the rate of growth in birds and their development of mature function. The literature was surveyed for data on growth and development, and the growth curves of 81 species were chosen for the analysis. Growth curves of most species were fitted with the Gompertz equation, and the rate constants of the equation were used as an index of the growth rate. For those species whose curves were fitted better by other equations, with a slightly different form, appropriate conversion factors, derived in this paper, were employed.
Among species with similar modes of development, growth rate decreases with increasing body weight in an allometric manner, with slopes of –0.26 to –0.42, depending on the group. Between groups, the rate of growth in body weight was found to be closely associated with the rate of development of function, in particular, the acquisition of flight. Among those species that can walk at an early age, but acquire flight relatively late, the rate of growth depends primarily on the relative size of the musculature of the lower extremities.
Data are presented to refute the hypotheses that growth rate is adjusted to nestling mortality, or that the energy requirements of the young (and hence their growth rates) are balanced against brood size. It is concluded that most species grow at some physiologically maximum rate, but as yet it is not possible to distinguish between limitation of growth rate at the level of the organism or at the level of the tissue.     

The effects of selection for different combinations of weights at two ages on the growth curve of mice

Summary The weights of mice in lines selected for different combinations of high and low body weights at 5 and at 10 weeks of age were recorded from 3 to 21 weeks of age. The average growth curve for each line was computed using the Gompertz function. The growth curves of lines selected for high or low weight at a single age (ST lines) showed large differences in estimates of mature size and small differences in estimates of maturing rate, i.e. of the relative rate of growth to maturity. The growth curves of lines selected by independent culling for divergent combinations of deviations of opposite sign in 5- and 10-week weights (ICL lines) showed little difference in estimates of mature size and a large difference in estimates of maturing rate. The growth curves of lines selected by index for divergence in 5-week weight with no change in 10-week-weight or for divergence in 10-week-weight with no change in 5-week weight showed large differences in estimates of mature size and large differences in estimates of the maturing rate. The relationship between mature size and maturing rate was affected in different ways by the three types of selection.     

Maturation patterns in female European eel: age and size at the silver eel stage

The maturation pattern in the female European eel Anguilla anguilla was studied by investigating age and size patterns of silver eels in different aquatic environments in Sweden, covering limnic, brackish and marine waters. The results neither supported the hypothesis that there is a critical size or age when eels enter the silvery stage, nor that size and age at maturity are positively related. Age at maturity, however, was observed to be negatively related to growth rate in all localities, i.e. the female reproductive tactic apparently is to become sexually mature at the earliest possible opportunity. Furthermore, it was recognized that a significant amount of variation was due to habitat differences, since the female eel maturation pattern deviated systematically between sampling sites, as it did also when the effect of growth rate was eliminated. Thus, the ability of the female eel to adjust maturation to an optimal size and age can be questioned, because the panmictic nature of the eel means local adaptations are unlikely Growth rate dependent differences suggest that variations in maturation patterns between eel environments are linked more to the opportunity for nutrient accumulation than to other aspects of growth.     

Optimal age and size at maturity in annuals and perennials with determinate growth

Summary A model predicting optimal age and size at maturity is presented, exploring the conflict between growth and energy allocation to reproduction. According to the model, the factors promoting delayed maturity and large adult body size are as follows: (1) high rate of somatic growth, (2) high percentage increase in reproductive rate with body size increase, (3) long life expectancy at maturity for annuals or large number of expected productive days (when either growth or reproduction is possible) for perennials with growth ceasing at maturity, (4) life expectancy increasing with body size. All these factors are combined in the mathematical formula predicting optimal age and size at maturity, which allows for quantitative predictions. The optimal schedule of growth and reproduction may be achieved by natural selection, developmental plasticity, or when one species replaces another. Sexual size dimorphism is also discussed, resulting from different optimal age at maturity for either sex.     

The analysis of reaction norms for age and size at maturity using maturation rate models

Reaction norms for age and size at maturity are being analyzed to answer important questions about the evolution of life histories. A new statistical method is developed in the framework of time-to-event data analysis, which circumvents shortcomings in currently available approaches. The method emphasizes the estimation of age- and size-dependent maturation rates. Individual probabilities of maturation during any given time interval follow by integrating maturation rate along the growth curve. The integration may be performed in different ways, over ages or sizes or both, corresponding to different assumptions on how individuals store the operational history of the maturation process. Data analysis amounts to fitting generalized nonlinear regression models to a maturation status variable. This technique has three main advantages over existing methods: (1) treating maturation as a stochastic process enables one to specify a rate of maturation; (2) age and size at which maturation occurs do not have to be observed exactly, and bias arising from approximations and interpolations is avoided; (3) ages at which sizes are measured and maturation status are observed can differ between individuals. An application to data on the springtail Folsomia candida is presented. Models with age-dependent integration of maturation rates were preferred. The analysis demonstrates a significant size dependence of the maturation rate but no age dependence.     

Ontogenetic variation in density‐dependent growth of brown trout through habitat competition

1. Density‐dependent growth has been widely reported in freshwater fishes, but the ontogenetic evolution of competition and its subsequent effects on growth through a life span remains unclear. 2. Patterns of competition can be described by integrating population abundance data with habitat‐modelling results. Weighted usable area (WUA; m² WUA ha^?1) curves are obtained for each flow value and are then coupled with demographic data to obtain the occupancy rates (trout m^?2 WUA, the density of a given age class related to its suitable habitat) of the WUA for every age class, year and site. 3. We examined a long‐term data series searching for temporal variation in the influence of habitat occupancy rate on the growth of brown trout Salmo trutta. We tested whether (i) mean cohort mass (mean mass of the cohort during the first 3 years of life) is affected by the occupancy rate experienced across a life span; and (ii) the occupancy rate experienced at different ages influenced mean body size. 4. We observed a consistent negative power relationship between average cohort mass and mean occupancy rate through a life span, indicating that stronger cohorts were related to a reduced growth, with likely consequences for individual fitness. 5. The effects of occupancy rate on size‐at‐age were mainly detected in the size attained at the second year of life, but they were because of the competition at different times. Thus, the level of competition varied through ontogeny, in some of the rivers affecting growth since the first year of life, whereas in most of the rivers the main effects on body size resulted from the competition during the second year of life. 6. Occupancy rate appears more appropriate than density for assessing the occurrence of habitat competition in freshwater fishes, since it encompasses the differences in quantity and quality of suitable habitat for each age class. 7. Our study highlights the importance of density‐dependent growth as a key process in the dynamics of brown trout populations, its temporal variation depending on the temporal changes of density and the variation of competition associated with the habitat capacity for each life stage.     

Determination of the unknown age at first capture of western rock lobsters (Panulirus cygnus) by random effects model

We propose a method for fitting growth curves to multiple recapture data of lobsters when the age at first capture is unknown. The von Bertalanffy growth curve is used to model the growth. To account for individual variability, the unknown age in logarithmic scale of a lobster at first capture, the individual asymptotic size, and the individual growth coefficient of its carapace length are modeled as random effects with a trivariate normal distribution. Unlike previously suggested models, the present model permits correlation between the growth coefficient and the age at first capture and can be fitted readily using existing software. The error structures between consecutive recaptures of a lobster are assumed to be a first-order autoregressive process with unequally spaced time points. A comparison between this model and the Fabens growth equation is given. The proposed method is a flexible method and can be applied to fit different growth equations when the age at first capture is unknown.     

The Ontogeny of Encephalization: Tradeoffs Between Brain Growth,Somatic Growth,and Life History in Hominoids and Platyrrhines

This study examines variation in brain growth relative somatic growth in four hominoids and three platyrrhines to determine whether there is a trade-off during ontogeny. I predicted that somatic growth would be reduced during periods of extensive brain growth, and species with larger degrees of encephalization would reach a smaller body size at brain growth completion because more energy is directed towards the brain. I measured cranial capacity and skeletal size in over 500 skeletal specimens from wild populations. I calculated nonlinear growth curves and velocity curves to determine brain/body growth allometry during ontogeny. In addition, I calculated linear regressions to describe the brain/body allometry during the postnatal period prior to brain size reaching an asymptote. The results showed that somatic growth is not substantially reduced in species with extensive brain growth, and body size at brain growth completion was larger in species with greater degrees of encephalization. Furthermore, large body size at brain growth completion was not correlated with interbirth interval, but was significantly correlated with prolonged juvenile periods and late age at maturity when data were corrected for phylogeny. These results indicate that neither reduction in body growth nor reproductive rate are compensatory mechanisms for the energetic costs of brain growth. Other avenues for meeting energetic costs must be in effect. In addition, the results show that somatic growth in encephalized species is particularly slow during the juvenile period after brain growth at or near completion, suggesting that these growth patterns are explained by reasons other than energetic costs.     

Modeling of body mass index by Newton's second law

Since laws of physics exists in nature, their possible relationship to terrestrial growth is introduced. By considering the human body as a dynamic system of variable mass (and volume), growing under a gravity field, it is shown how natural laws may influence the vertical growth of humans. This approach makes sense because the non-linear percentile curves of different aspects of human physical growth from childhood to adolescence can be described in relation to physics laws independently of gender and nationality. Analytical relations for the dependence of stature, measured mass (weight), growth velocity (and their mix as the body mass index) on age are deduced with a set of common statistical parameters which could relate environmental, genetics and metabolism and different aspects of physical growth on earth. A relationship to the monotone smoothing using functional data analysis to estimate growth curves and its derivatives is established. A preliminary discussion is also presented on horizontal growth in an essentially weightless environment (i.e., aquatic) with a connection to the Laird-Gompertz formula for growth.     

The record of daily growth in otoliths of Atlantic silversides, Menidia menidia, from field and laboratory

Otoliths were removed from field-collected silversides of age less than 3 months. Otolith diameter was highly correlated with total length of the fish. Daily growth ring counts for this species are known to be a function of age rather than size, so widths for the daily growth rings provide a record of daily increases in length of the fish. Measurement of ring widths showed that weekly specific growth rate was greater than 70% at age 1 week, but declined to about 30% at age 1 month and about 15% at age 2 months. A laboratory experiment in which temperature was changed on a weekly basis demonstrated that environmental variables can affect the width of rings. Nevertheless, the growth rate of field-collected fish, as calculated from otolith ring widths, was more highly correlated with size of fish, as measured by otolith radius, than with the environmental variables of temperature, salinity and plankton abundance. Back-calculation of growth rates from otolith ring widths of five fish collected at the end of the growing season yielded the same age-growth curves as were obtained from 203 fish collected biweekly during the season.     

Cell population dynamics during the differentiative phase of tissue development

The dynamics of a cell population whose numbers are growing exponentially have been described well by a mathematical model based on the theory of age-dependent branching processes. Such a model, however, does not cover the period following exponential growth when cell differentiation curtails population size. This paper offers an extension to the branching process model to remedy this deficiency. The extended model is ideal for describing embryonic growth; its use is illustrated with data from embryonic retina. The model offers a better computational framework for the interpretation of a variety of data (growth curves of cell numbers, DNA histograms, thymidine labelling indices, FLM curves, BUdR-labelled mitoses curves) because age-distributions can be calculated at any stage of development, not just during exponential growth. Proportions of cells in the various phases of the cell cycle can be computed as growth slows. Such calculations show the gradual transition from a population dominated by cells which are young with respect to cell cycle age to one dominated by those which are old, and the effects such biases have on the proportions of cells in each phase.     

Analytical model of hygroscopic particle behavior in human airways

A model is developed to calculate the deposition of hygroscopic aerosols in the human tracheobronchial (TB) tree. The TB airflow pattern assumed is consistent with experimental observations and accounts for anatomical features such as the larynx and cartilaginous rings in large airways. Some original deposition efficiency formulae are presented for laminar and turbulent airstreams. Stepwise growth is simulated by changes in particle size and density at each TB generation. The dose distribution of NaCl aerosols is studied as a function of inhaled particle size and flow rate. Two NaCl growth rate curves are used which differ in the mode of aerosol-air mixing in the trachea. The initial rate of aerosol mixing in the human due to the laryngeal jet is shown to be an important factor affecting the deposition of hygroscopic aerosols. Total TB deposition of NaCl exceeds that for nonhygroscopic particles of the same inhaled aerodynamic size. Hygroscopic growth can also influence the regional TB distribution of dose when submicron NaCl particles grow rapidly enough to deposit by impaction and sedimentation.     

Biased estimates of fur seal pup mass: origins and implications

The mass of fur seal pups weighed in different years can be used to estimate growth rates or compared with one another to make inferences about the relative condition of a population. However, unless appropriate precautions are taken, many factors can bias estimates of pup mass and lead to incorrect conclusions. Using data collected from tagged and untagged northern fur seal pups ( Callorhinus ursinus ) at the Pribilof Islands, Alaska, I assess how milk consumption, the timing of sampling, and the effects of growth and sample size influence the size of pups captured for weighing. Evidence is presented suggesting that pup mass may increase in a sigmoid fashion, with the most rapid rate of growth occurring when about two months old. This phenomenon can confound efforts to compare the masses of pups weighed on different days in different years, particularly if pups are weighed over the period of rapid growth. Variability in pup mass increases with time because growth rates of individuals vary and because the amount of milk pups consume increases with body size. Thus sample sizes must be increased as the pups grow older in order to detect statistically significant differences in mean body mass. There is also evidence that pups of different ages and sizes are not randomly distributed on the breeding beaches and are not randomly selected for weighing. It appears that the first pups captured for weighing are smaller and younger than subsequent captures, possibly because smaller pups are easier to handle and are segregated to the peripheral rookery regions where sampling begins. These hidden biases, related to sampling error and fur seal biology, must be considered and controlled for when weighing fur seal pups.     

A New Generalized Logistic Sigmoid Growth Equation Compared with the Richards Growth Equation

A new sigmoid growth equation is presented for curve-fitting,analysis and simulation of growth curves. Like the logisticgrowth equation, it increases monotonically, with both upperand lower asymptotes. Like the Richards growth equation, itcan have its maximum slope at any value between its minimumand maximum. The new sigmoid equation is unique because it alwaystends towards exponential growth at small sizes or low densities,unlike the Richards equation, which only has this characteristicin part of its range. The new sigmoid equation is thereforeuniquely suitable for circumstances in which growth at smallsizes or low densities is expected to be approximately exponential,and the maximum slope of the growth curve can be at any value.Eleven widely different sigmoid curves were constructed withan exponential form at low values, using an independent algorithm.Sets of 100 variations of sequences of 20 points along eachcurve were created by adding random errors. In general, thenew sigmoid equation fitted the sequences of points as closelyas the original curves that they were generated from. The newsigmoid equation always gave closer fits and more accurate estimatesof the characteristics of the 11 original sigmoid curves thanthe Richards equation. The Richards equation could not estimatethe maximum intrinsic rate of increase (relative growth rate)of several of the curves. Both equations tended to estimatethat points of inflexion were closer to half the maximum sizethan was actually the case; the Richards equation underestimatedasymmetry by more than the new sigmoid equation. When the twoequations were compared by fitting to the example dataset thatwas used in the original presentation of the Richards growthequation, both equations gave good fits. The Richards equationis sometimes suitable for growth processes that may or may notbe close to exponential during initial growth. The new sigmoidis more suitable when initial growth is believed to be generallyclose to exponential, when estimates of maximum relative growthrate are required, or for generic growth simulations.Copyright1999 Annals of Botany Company Asymptote,Cucumis melo,curve-fitting, exponential growth, intrinsic rate of increase, logistic equation, maximum growth rate, model, non-linear least-squares regression, numerical algorithm, point of inflexion, relative growth rate, Richards growth equation, sigmoid growth curve.     

The effects of temperature and ultraviolet irradiation on multiplication of bacteriophage phi29.

The effects of temperature and of ultraviolet radiation on the multiplication of bacteriophage phi29 were studied. Samples of phi29 that had been irradiated to surviving fractions of 0.44 or 0.10 were propagated at 37 degrees C, 42 degrees C and 43.5 degrees C. Latent periods and burst sizes were obtained from one-step growth curves. At a particular temperature, as the dose delivered to the virus was increased, the latent period was extended and the burst size was decreased. For unirradiated virus, the burst size was the same at 42 degrees C as at 37 degrees C, but decreased dramatically at 43.5 degrees C. For virus subjected to a particular dose, the burst size decreased as the temperature was raised. A statistical technique for improving the reliability of parameters obtained from one-step growth curves is presented.     

Optimal individual growth and reproduction in perennial species with indeterminate growth

Summary A model predicting optimal timing of growth and reproduction in perennial species with indeterminate growth living in a seasonal environment, is presented. According to the model, the optimal fraction of growing season devoted to growth decreases with increasing individual age and size, which leads to S-shaped growth curves. Winter mortality seems to be a crucial factor affecting the timing of growth and reproduction, under the same function describing the dependence of growth rate and reproductive rate on body size. When winter mortality is heavy, it is often optimal to start reproducing in the first year, and to devote a large proportion of the subsequent years to reproduction, thus leading to small adult body sizes.The model has been applied to two species of mollusc and one species of fish. The model predictions fit well to the field data for these three species.     

Evaluation of the growth rate of MCF-7 breast cancer multicellular spheroids using three mathematical models

Abstract. Growth data on 60 multicellular spheroids of MCF-7 human breast cancer cells were fitted, on an individual basis, by the Gompertz, Bertalanffy and logistic equations. MCF-7 spheroids, initiated and grown in medium containing oestrogens, exhibited a growth rate that decreased continuously as spheroid size increased. Plots of spheroid volume v. time generated sigmoid curves that showed an early portion with an approximately exponential volume increase; a middle region or retardation phase characterized by a continuously decreasing growth rate; and, finally, a late segment or plateau phase approaching zero growth rate, that permitted an estimate of the maximum spheroid size (V_max). Growth curves generated by MCF-7 spheroids under different experimental conditions (hormones, drugs and radiation exposures) can be compared after normalization. Linearized forms of the fitted Gompertz curves provided a convenient way to express differences in growth rate.     

Using otolith weight to age fish

The problem of determining and verifying ages of fish, from populations having a considerable variation in size at age, has been investigated using the relationship between otolith size and fish size, which has been shown by several authors to be influenced by growth rate. In such a population of Sardinella aurita Val. an index of age can be obtained for individual fish by calculating the equivalent otolith weight at a particular fish length, using the otolith weight–fish length relationship determined for each age group. This statistic not only permits a much greater proportion of fish to be assigned ages than is possible with otolith reading alone, but also enables the age groups to be verified as year classes. However, it is concluded that, although appropriate models based on otolith-fish size relationships can predict age for groups of fish in which growth rates are known or can be assumed to be consistent, such techniques have a limited application in ageing fish from wild populations with highly variable growth rates.