A unified approach to the Richards-model family for use in growth analyses: Why we need only two model forms

This paper advances a unified approach to the modeling of sigmoid organismal growth. There are numerous studies on growth, and there have been several proposals and applications of candidate models. Still, a lack of interpretation of the parameter values persists and, consequently, differences in growth patterns have riddled this field. A candidate regression model as a tool should be able to assess and compare growth-curve shapes, systematically and precisely. The Richards models constitute a useful family of growth models that amongst a multitude of parameterizations, re-parameterizations and special cases, include familiar models such as the negative exponential, the logistic, the Bertalanffy and the Gompertz. We have reviewed and systemized this family of models. We demonstrate that two specific parameterizations (or re-parameterizations) of the Richards model are able to substitute, and thus to unify all other forms and models. This unified-Richards model (with its two forms) constitutes a powerful tool for an interpretation of important characteristics of observed growth patterns, namely, [I] maximum (relative) growth rate (i.e., slope at inflection), [II] age at maximum growth rate (i.e., time at inflection), [III] relative mass or length at maximum growth rate (i.e., relative value at an inflection), [IV] value at age zero (i.e., birth, hatching or germination), and [V] asymptotic value (i.e., adult weight or length). These five parameters can characterize uniquely any sigmoid-growth data. To date most studies only compare what is referred to as the “growth-rate constant” or simply “growth rate” (k). This parameter can be interpreted as neither relative nor actual growth rate, but only as a parameter that affects the slope at inflection. We fitted the unified-Richards and five other candidate models to six artificial data sets, generated from the same models, and made a comparison based on the corrected Akaike’s Information Criterion (AICc). The outcome may in part be the result of the random generation of data points. Still, in conclusion, the unified-Richards model performed consistently well for all data sets, despite the penalty imposed by the AICc.     

Growth and annual survival estimates to examine the ecology of larval lamprey and the implications of ageing error in fitting models

This study used existing western brook lamprey Lampetra richardsoni age information to fit three different growth models (i.e. von Bertalanffy, Gompertz and logistic) with and without error in age estimates. Among these growth models, there was greater support for the logistic and Gompertz models than the von Bertalanffy model, regardless of ageing error assumptions. The von Bertalanffy model, however, appeared to fit the data well enough to permit survival estimates; using length‐based estimators, annual survival varied between 0·64 (95% credibility interval: 0·44–0·79) and 0·81 (0·79–0·83) depending on ageing and growth process error structure. These estimates are applicable to conservation and management of L. richardsoni and other western lampreys (e.g. Pacific lamprey Entosphenus tridentatus) and can potentially be used in the development of life‐cycle models for these species. These results also suggest that estimators derived from von Bertalanffy growth models should be interpreted with caution if there is high uncertainty in age estimates.     

长江干流季节性水淹滩地杨树生长规律研究

研究了长江中下游干流滩地抑螺防病林的主要树种杨树的生长规律,建立了杨树的生 长动态数学模型．经过进一步的生长动态模型分析,研究结果表明：滩地高程是影响杨树胸径、 树高和材积生长量的关键因子．滩地淹水时间超过６０天的地方,杨树生长受到明显的影响．     

Age and growth of the sandbar shark, Carcharhinus plumbeus, in Hawaiian waters through vertebral analysis

Age and growth estimates were determined for the sandbar shark, Carcharhinus plumbeus, from Oahu, Hawaii in the central Pacific Ocean. Age estimates were obtained through vertebral centra analysis of 187 sharks. We verified our age estimates through marginal increment analysis of centra and oxytetracycline marking methods of at liberty sandbar sharks. Sizes of sampled sharks ranged from 46 to 147 cm pre-caudal length. Four growth models were fitted to length-at-age data; two forms of the von Bertalanffy growth model, the Gompertz growth model, and a logistic growth model. Males and females exhibited statistically significant differences in growth, indicating that females grow slower and attain larger sizes than males. Growth parameter estimates revealed slower growth rates than previously estimated (based on captive specimens) for Hawaiian sandbar sharks. The von Bertalanffy growth model using empirical length-at-birth provided the best biological and statistical fit to the data. This model gave parameter estimates of L _∞ = 138.5 cm PCL and k = 0.12 year⁻¹ for males and L _∞ = 152.8 cm PCL, k = 0.10 year⁻¹ for females. Male and female sandbar sharks mature at approximately 8 and 10 years of age, respectively.     

Temperature and salinity effects on post-marsupial growth of Neomysis integer (Crustacea: Mysidacea)

There has been an increasing interest in using the brackish water mysid Neomysis integer as a toxicological test species for Western European estuarine systems. In this respect, more data on growth, moulting and development in this species is needed. The influence of prevailing environmental variables (e.g. temperature, salinity) and age on these processes as well as their optimal range have to be known in order to develop optimal laboratory cultures and to differentiate between chemically induced variability and natural variability in toxicity testing. Individual post-marsupial growth (size, intermoult period, growth factor) was studied from first day neonates until adulthood at eight environmentally relevant temperature-salinity conditions. Three salinities (5, 15 and 30 psu) were tested at 15 and 20 °C, and two more extreme temperatures (8 and 25 °C) were tested at a salinity of 5 psu.Survival and growth of N. integer were detected within the whole range tested, but sexual maturation was only possible in the narrower range of 15-25 °C and 5-15 psu. The size at maturity of N. integer increased with decreasing temperature and increasing salinity. Salinity seems to have a stronger effect than temperature on the duration of maturation. The sigmoid von Bertalanffy growth model was fitted to the individual and pooled data, except for the 8 °C experiment where growth was linear. Estimates from pooled data were comparable with individually based estimates, but generally underestimated the asymptotic length. Temperature was negatively correlated with the asymptotic length and positively correlated with the growth constant K. Higher temperatures caused smaller intermoult periods but had no effect on the growth increment, while salinity effects were less straightforward and dependent on the water temperature. A tool is provided to estimate the age, moult number, intermoult period, growth factor and growth rate from the body standard length of N. integer. Experimentally derived von Bertalanffy parameter estimates resulted in a higher growth performance index compared with field-based estimates for the Schelde estuary and Galgenweel populations of N. integer.     

Modeling of microbial substrate conversion,growth and product formation in a recycling fermentor

Paracoccus denitrificans and Bacillus licheniformis were grown in a carbon- and energy source-limited recycling fermentor with 100% biomass feedback. Experimental data for biomass accumulation and product formation as well as rates of carbon dioxide evolution and oxygen consumption were used in a parameter optimization procedure. This procedure was applied on a model which describes biomass growth as a linear function of the substrate consumption rate and the rate of product formation as a linear function of the biomass growth rate. The fitting procedure yielded two growth domains for P. denitrificans. In the first domain the values for the maximal growth yield and the maintenance coefficient were identical to those found in a series of chemostat experiments. The second domain could be described best with linear biomass increase, which is equal to a constant growth yield. Experimental data of a protease producing B. licheniformis also yielded two growth domains via the fitting procedure. Again, in the first domain, maximal growth yield and maintenance requirements were not significantly different from those derived from a series of chemostat experiments. Domain 2 behaviour was different from that observed with P. denitrificans. Product formation halts and more glucose becomes available for biomass formation, and consequently the specific growth rate increases in the shift from domain 1 to 2. It is concluded that for many industrial production processes, it is important to select organisms on the basis of a low maintenance coefficient and a high basic production of the desired product. It seems less important that the maximal production becomes optimized, which is the basis of most selection procedures.     

Analysis of growth patterns in purebred Kambing Katjang goat and its crosses with the German Fawn

The objective of this study was to investigate growth patterns of goats utilizing data from a crossbreeding program involving the exotic German Fawn (GF) and the indigenous Kambing Katjang (KK) goats. Growth curve models and growth curve parameters were compared and analyzed for different genotypes and litter types. A total of 20,393 weight–age data from 208 female goats belonging to various crossbreeding genotypes were individually fitted to four growth curve models (Brody, Bertalanffy, Gompertz and Logistic). The goodness of fit was highest in the Brody model in most cases. A comparison of R² among genotypes showed that they were highest for KK. There were no significant differences of genotypes for estimated mature weight in the Brody model. The estimated mature weights for KK were significantly lower (P < 0.05) than for GF × KK (F₁), backcrosses with 75% GF genes (BC) and F₁ × F₁ (F₂) in the other models. The correlations between estimated mature weights and the maturing rates were lowest for BC. The genotype significantly (P < 0.01) affected the age at the constant degree of maturity (67% and 90% of mature weight) in all models. The BC genotype was the youngest at maturity and KK the oldest. All models well expressed the growth pattern of the target animals when they were older than 2.5 years of age. The results from the present study showed that the growth pattern may be altered by crossbreeding of KK with the GF breed.     

Age,growth, maturity and extinction risk of an exploited and endangered skate,Atlantoraja castelnaui,from off Uruguay and northern Argentina

The spotback skate Atlantoraja castelnaui (Arhynchobatidae) is a large and threatened skate species subjected to fishing pressure, endemic to the Southwest Atlantic that occurs from Rio de Janeiro, Brazil, to San Jorge Gulf, Argentina. The age, growth, age at maturity and the maximum intrinsic rate of population increase r_max of A. castelnaui were studied using 152 specimens collected from off Uruguay and north Argentina (35°–42° S), between June 2013 and February 2020. Vertebrae from 143 individuals were used for ageing (females: n = 83, size range 404–1300 mm total length, TL; males: n = 60, size range 400–1270 mm TL). Maximum ages determined for females and males were 30 and 28 years, respectively. To fit growth models, non-linear and Bayesian estimation approaches were considered. For the first approach, a set of four candidate growth (size-at-age) models were fitted: three-parameter von Bertalanffy, two-parameter von Bertalanffy with fixed L₀, Gompertz and Logistic. In the second approach, von Bertalanffy, Gompertz and Logistic were fitted. For non-linear estimation, model selection indicated that the entire set of candidate growth models were supported by the data. The von Bertalanffy was selected as the best model for Bayesian estimation. There were no differences in growth between sexes. For the sexes combined, the von Bertalanffy growth model by Bayesian method was considered the most adequate to describe the growth of A. castelnaui (growth mean parameters ± S.D. : L_∞ = 1210.29 ± 40.68 mm; k = 0.12 ± 0.01 years⁻¹; L₀ = 179.20 ± 11.62 mm). The age at maturity was estimated at 16.21 and 14.04 years for females and males, respectively. The maximum intrinsic rate of population increase r_max was estimated as 0.252 years⁻¹. Life-history traits and r_max provided in the present study suggest that this species has a relatively low productivity and may be vulnerable to an intense fishing pressure.     

Development of sexual dimorphism in two sympatric skinks with different growth rates

Sexual size dimorphism (SSD) is widespread in animals, especially in lizards (Reptilia: Squamata), and is driven by fecundity selection, male–male competition, or other adaptive hypotheses. However, these selective pressures may vary through different life history periods; thus, it is essential to assess the relationship between growth and SSD. In this study, we tracked SSD dynamics between a “fading‐tail color skink” (blue tail skink whose tail is only blue during its juvenile stage: Plestiodon elegans) and a “nonfade color” tail skink (retains a blue tail throughout life: Plestiodon quadrilineatus) under a controlled experimental environment. We fitted growth curves of morphological traits (body mass, SVL, and TL) using three growth models (Logistic, Gompertz, and von Bertalanffy). We found that both skinks have male‐biased SSD as adults. Body mass has a higher goodness of fit (as represented by very high R² values) using the von Bertalanffy model than the other two models. In contrast, SVL and TL for both skinks had higher goodness of fit when using the Gompertz model. Two lizards displayed divergent life history tactics: P. elegans grows faster, matures earlier (at 65 weeks), and presents an allometric growth rate, whereas P. quadrilineatus grows slower, matures later (at 106 weeks), and presents an isometric growth rate. Our findings imply that species‐ and sex‐specific trade‐offs in the allocation of energy to growth and reproduction may cause the growth patterns to diverge, ultimately resulting in the dissimilar patterns of SSD.     

Richards方程的数学属性及其在兽类生长过程中的应用

通过对Richards方程数学属性的分析表明 ,该方程具有变动的拐点值 ,因而在描绘兽类多种多样的生长过程时具有良好的可塑性。依据其方程参数n取值的不同 ,Richards方程包含了Spillman ,Logistic,Gompertz以及Bertalanffy方程。为了评估Richards方程对兽类生长过程的拟合优度 ,作者引用 1 0组哺乳动物兽类生长数据 ,将它与一些经典的生长模型如Spillman ,Logistic,Gompertz以及Bertalanffy方程共同进行了拟合比较。结果表明 ,Richards方程具有良好的拟合优度 ,适于描绘多种多样的兽类生长模式。     

The effects of flow and stream characteristics on the variation in freshwater mussel growth in a Southeast US river basin

相似文献   

Morphological changes and resource allocation of Zizania latifolia (Griseb.) Stapf in response to different submergence depth and duration

Plants around ponds, rivers and lakes are subjected to long-term partial or complete submergence. When they are flooded, water level affects the plants simultaneously with duration of submergence. Separate and interactive effects of water level and duration on the growth of the herbaceous perennial Zizania latifolia (Poaceae) were investigated by exposing the plants in greenhouse water tanks to submergence in different water depths and for different time-spans. The plants exhibited great shoot elongation upon submergence and prolonged flood duration, and the basal tiller number of the species decreased with higher water levels. Submergence treatment advanced the flowering date and increased the inflorescence number. Plant total biomass did not differ among all the treatments, while the root:shoot ratio decreased with increased water level, prolonged duration of submergence and their interaction. The high plasticity in morphology and shifts in reproductive strategy and biomass allocation enabled the Zizania plants to survive the compound effect of flooding height and duration. This may explain the occurrence of this species in habitats subjected to long-term flooding. The results obtained in this experiment will contribute to understanding the impact of flooding dynamics on plants and the ways of adaptation responses to prolonged waterlogging.     

Experimental validation of in silico estimated biomass yields of Pseudomonas putida KT2440

Pseudomonas putida is rapidly becoming a microbial cell platform for biotechnological applications. In order to understand genotype‐phenotype relationships genome scale models represent helpful tools. However, the validation of in silico predictions of genome scale models is a task that is rarely performed. In this study the theoretical biomass yields of Pseudomonas putida KT2440 were estimated for 57 different carbon sources based on a genome scale stoichiometric model applying flux balance analysis. The batch growth of P. putida KT2440 with six individual carbon sources covering the range of maximal to minimal in silico biomass yields (acetate, glycerol, citrate, succinate, malate and methanol, respectively) was studied in a defined mineral medium in a fully controlled stirred‐tank bioreactor on a 3 L scale. The highest growth rate of P. putida KT2440 was measured with succinate as carbon source (0.51 h^?1). Among the 57 carbon sources tested, glycerol resulted in the highest estimated biomass yield (0.61 molC_Biomass molC^?1_Glycerol) which was experimentally confirmed. The comparison of experimental determined biomass yields with a modified version of the model iJP815 showed deviations of only up to 10%. The experimental data generated in this study can also be used in future studies to further improve the genome scale models of P. putida KT2440. Improved models will then help to gain deeper insights in genotype‐phenotype relationships.     

Reproductive allocation of an annual,<Emphasis Type="Italic"> Xanthium canadense</Emphasis>, at an elevated carbon dioxide concentration

Stimulation of vegetative growth by an elevated CO₂ concentration does not always lead to an increase in reproductive yield. This is because reproductive yield is determined by the fraction of biomass allocated to the reproductive part as well as biomass production. We grew Xanthium canadense at low N (LN) and high N levels (HN) under an ambient (360 mol mol^-1) and elevated (700 mol mol^-1) CO₂ concentration ([CO₂]) in open-top chambers. Reproductive yield was analysed as the product of: (1) the duration of the reproductive period, (2) the rate of dry mass acquisition in the reproductive period, and (3) the fraction of acquired biomass allocated to the reproductive part. Elevated [CO₂] increased the total amount of biomass that was allocated to reproductive structures, but this increase was caused by increased capsule mass without a significant increase in seed production. The increase in total reproductive mass was due mainly to an increase in the rate of dry mass acquisition in the reproductive period with a delay in leaf senescence. This positive effect was partly offset by a reduction in biomass allocation to the reproductive part at elevated [CO₂] and HN. The duration of the reproductive period was not affected by elevated [CO₂] but increased by HN. Seed production was strongly constrained by the availability of N for seed growth. The seed [N] was very high in X. canadense and did not decrease significantly at elevated [CO₂]. HN increased seed [N] without a significant increase in seed biomass production. Limited seed growth caused a reduction in biomass allocation to the reproductive part even though dry mass production was increased due to increased [CO₂] and N availability.     

Comparative analysis of nonlinear growth curve models for <Emphasis Type="Italic">Arabidopsis thaliana</Emphasis> rosette leaves

As a model organism, modeling and analysis of the phenotype of Arabidopsis thaliana (A. thaliana) leaves for a given genotype can help us better understand leaf growth regulation. A. thaliana leaves growth trajectories are to be nonlinear and the leaves contribute most to the above-ground biomass. Therefore, analysis of their change regulation and development of nonlinear growth models can better understand the phenotypic characteristics of leaves (e.g., leaf size) at different growth stages. In this study, every individual leaf size of A. thaliana rosette leaves was measured during their whole life cycle using non-destructive imaging measurement. And three growth models (Gompertz model, logistic model and Von Bertalanffy model) were analyzed to quantify the rosette leaves growth process of A. thaliana. Both graphical (plots of standardized residuals) and numerical measures (AIC, R² and RMSE) were used to evaluate the fitted models. The results showed that the logistic model fitted better in describing the growth of A. thaliana leaves compared to Gompertz model and Von Bertalanffy model, as it gave higher R² and lower AIC and RMSE for the leaves of A. thaliana at different growth stages (i.e., early leaf, mid-term leaf and late leaf).     

Ozone effects on wheat in relation to CO2: modelling short‐term and long‐term responses of leaf photosynthesis and leaf duration

A combined stomatal–photosynthesis model was extended to simulate the effects of ozone exposure on leaf photosynthesis and leaf duration in relation to CO₂. We assume that ozone has a short‐term and a long‐term effect on the Rubisco‐limited rate of photosynthesis, A_c. Elevated CO₂ counteracts ozone damage via stomatal closure. Ozone is detoxified at uptake rates below a threshold value above which A_c decreases linearly with the rate of ozone uptake. Reduction in A_c is transient and depends on leaf age. Leaf duration decreases depending on accumulated ozone uptake. This approach is introduced into the mechanistic crop simulation model AFRCWHEAT2. The derived model, AFRCWHEAT2‐O3, is used to test the capability of these assumptions to explain responses at the plant and crop level. Simulations of short‐term and long‐term responses of leaf photosynthesis, leaf duration and plant and crop growth to ozone exposure in response to CO₂ are analysed and compared with experimental data derived from the literature. The model successfully reproduced published responses of leaf photosynthesis, leaf duration, radiation use efficiency and final biomass of wheat to elevated ozone and CO₂. However, simulations were unsatisfactory for cumulative radiation interception which had some impact on the accuracy of predictions of final biomass. There were responses of leaf‐area index to CO₂ and ozone as a result of effects on tillering which were not accounted for in the present model. We suggest that some model assumptions need to be tested, or analysed further to improve the mechanistic understanding of the combined effects of changes in ozone and CO₂ concentrations on leaf photosynthesis and senescence. We conclude that research is particularly needed to improve the understanding of leaf‐area dynamics in response to ozone exposure and elevated CO₂.     

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     

Age and growth of the smooth hammerhead,Sphyrna zygaena,in the Atlantic Ocean: comparison with other hammerhead species

The smooth hammerhead Sphyrna zygaena (Sphyrnidae) is a pelagic shark occasionally caught as bycatch in pelagic longline fisheries, but is one of the least studied of all pelagic sharks. Age and growth of S. zygaena was studied along a wide Atlantic region covering both the northern and southern hemispheres. Data from 304 specimens, caught between October 2009 and September 2014, ranging in size from 126 to 253?cm fork length (FL), were analysed. Growth models were fitted using the three-parameter von Bertalanffy growth function (VBGF) re-parameterized to calculate L₀ (size at birth). Growth models were fitted to the sample data and data from several back-calculation models. The model fit to the quadratic modified Dahl-Lea back-calculated data seems to be the most appropriate to describe growth in this species, with resulting growth parameters of L_inf?=?285?cm FL, k?=?0.09 year^?1 for males and L_inf?=?293?cm FL, k?=?0.09 year^?1 for females. Compared with other species of the same genus, estimated growth coefficients for S. zygaena seem to fall in the low to middle range. Although further work is still needed, this study adds to knowledge of the vital life-history parameters of smooth hammerheads in the Atlantic Ocean, which can be used in the management and conservation of this species.