Age is not just a number: How incorrect ageing impacts close-kin mark-recapture estimates of population size.

Population size is a key parameter for the conservation of animal species. Close-kin mark-recapture (CKMR) relies on the observed frequency and type of kinship among individuals sampled from the population to estimate population size. Knowledge of the age of the individuals, or a surrogate thereof, is essential for inference with acceptable precision. One common approach, particularly in fish studies, is to measure animal length and infer age using an assumed age-length relationship (a 'growth curve'). We used simulation to test the effect of misspecifying the length measurement error and the growth curve on population size estimation. Simulated populations represented two fictional shark species, one with a relatively simple life history and the other with a more complex life history based on the grey reef shark (Carcharhinus amblyrhynchos). We estimated sex-specific adult abundance, which we assumed to be constant in time. We observed small median biases in these estimates ranging from 1.35% to 2.79% when specifying the correct measurement error standard deviation and growth curve. CI coverage was adequate whenever the growth curve was correctly specified. Introducing error via misspecified growth curves resulted in changes in the magnitude of the estimated adult population, where underestimating age negatively biased the abundance estimates. Over- and underestimating the standard deviation of length measurement error did not introduce a bias and had negligible effect on the variance in the estimates. Our findings show that assuming an incorrect standard deviation of length measurement error has little effect on estimation, but having an accurate growth curve is crucial for CKMR whenever ageing is based on length measurements. If ageing could be biased, researchers should be cautious when interpreting CKMR results and consider the potential biases arising from inaccurate age inference.

Application of growth models to South African Boer goat castrates and does under feedlot conditions.

Mathematical models may aid researchers in describing biological processes, like growth, in animals. This study aimed to collect the body weight data of 18 Boer goat castrates and 20 Boer goat does, from birth until maturity, to model growth and determine growth trends. This is a novel investigation as sufficient information on an age-weight database for these two Boer goat sexes from birth to maturity, is lacking. Using age-weight data, four nonlinear models, namely the Brody, Gompertz, Logistic and Von Bertalanffy growth models, were plotted and evaluated. The model parameters of each growth model were compared for differences between the two sexes. The statistical effectiveness of fit was determined for each model using AIC and RMSE, with R2 also being considered. All models except the Brody model, predicted significantly heavier mature weights for castrates. The Brody model was deemed unfit to describe Boer goat growth as the function severely over-predict weights from birth until maturity for both sexes. The Von Bertalanffy (R2 = 91.3) and Gompertz functions (R2 = 91.3) showed the best fit for Boer goat castrates, while the Gompertz model (R2 = 95.1) showed the best fit for Boer goat does. The Gompertz function is the preferred model to depict Boer goat growth overall, as it accurately characterized growth of both sexes. According to the Gompertz model the age at which the inflection point of the growth curve was reached, did not differ significantly between castrates and does (141.80 days versus 136.31 days). There was also no significant difference in maturation rate between the two sexes.

Ulam type stability for von Bertalanffy growth model with Allee effect.

In many studies dealing with mathematical models, the subject is examining the fitting between actual data and the solution of the mathematical model by applying statistical processing. However, if there is a solution that fluctuates greatly due to a small perturbation, it is expected that there will be a large difference between the actual phenomenon and the solution of the mathematical model, even in a short time span. In this study, we address this concern by considering Ulam stability, which is a concept that guarantees that a solution to an unperturbed equation exists near the solution to an equation with bounded perturbations. Although it is known that Ulam stability is guaranteed for the standard von Bertalanffy growth model, it remains unsolved for a model containing the Allee effect. This paper investigates the Ulam stability of a von Bertalanffy growth model with the Allee effect. In a sense, we obtain results that correspond to conditions of the Allee effect being very small or very large. In particular, a more preferable Ulam constant than the existing result for the standard von Bertalanffy growth model, is obtained as the Allee effect approaches zero. In other words, this paper even improves the proof of the result in the absence of the Allee effect. By guaranteeing the Ulam stability of the von Bertalanffy growth model with Allee effect, the stability of the model itself is guaranteed, and, even if a small perturbation is added, it becomes clear that even a small perturbation does not have a large effect on the solutions. Several examples and numerical simulations are presented to illustrate the obtained results.

Genetic parameters of sheep growth curve traits reared within rangelands under uncertain paternity.

This work aims to improve the selection program of the Timahdit breed through the use of the parameters of the Von Bertalanffy model as selection criteria and the treatment of uncertain paternity found in pastoral systems. A database containing 12,029 animals and a pedigree file integrating the probabilities of the parents with a total of 48,292 animals were used in the analysis. An individual estimation of the parameters of the model studied by the nonlinear regression procedure Proc NLIN of SAS was carried out, followed by the determination of the fixed effects which influence these parameters by means of a general linear model using the GLM procedure of the SAS software. The treatment of uncertain paternity is solved by an R code translating Average Numerator Relationship Matrix Model (ANRM). Then, the variance and (co)variance components were estimated by a Bayesian approach using the BRMS package. The high heritability values obtained, between 0.52 and 0.55 for the parameters studied, suggest good prospects for genetic responses to selection and the maintenance of sustained genetic progress, especially when environmental conditions are unfavorable. The positive correlations between all the parameters studied show that animals with rapid development tend to have lower weight performance. Finally, with optimal selection based on the genetic values associated with these parameters, we can make the desired changes to the growth curve by choosing breeders that achieve high weight performances as quickly as possible, and that would allow improving the feed efficiency for these animals, as well as increasing the profitability of sheep farms.

A systematic review of estimation of growth curve in goats.

Growth is an economically important trait in animal production industry and is one of the subjects that can be justified mathematically. The literature recommends different non-linear model to estimate the growth of goats. The objective of this study was to systematically review the literature published on estimation of growth using non-linear models in goats. Databases such as Google Scholar, PubMed, ScienceDirect, and Web of Science were evaluated systematically using the combination of the following key terms: Non-linear growth curve models such as Brody, Richards, Gompertz, Von Bertalanffy, Logistic models. A total of 25 eligible articles were found published between 2008 and 2022 in Bangladesh, Brazil, China, Egypt, Germany, India, Indonesia, Iran, Pakistan, South Africa, Turkey, Tunisia, and Vietnam. The results showed that out of 25 articles, Gompertz growth curve model was the most used (n = 10), followed by Logistic (n = 8), then Brody growth curve model (n = 6). The findings further indicated that Janoscheck growth curve model was the least used model (n = 1) for estimation of growth in goats. One of the limitations is that some of the reviewed articles did not indicate the sex of the animals which make it difficult to draw the conclude for sexes. The systematic review concludes that Gompertz growth curve model is the most recommended for estimation of growth parameters of goats, followed by Logistic, and then Brody. Therefore, researchers should consider using these models when studying growth parameters of goats.

Downsized: gray whales using an alternative foraging ground have smaller morphology.

Describing individual morphology and growth is key for identifying ecological niches and monitoring the health and fitness of populations. Eastern North Pacific ((ENP), approximately 16 650 individuals) gray whales primarily feed in the Arctic/sub-Arctic regions, while a small subgroup called the Pacific Coast Feeding Group (PCFG, approximately 212 individuals) instead feeds between northern California, USA and British Columbia, Canada. Evidence suggests PCFG whales have lower body condition than ENP whales. Here we investigate morphological differences (length, skull, and fluke span) and compare length-at-age growth curves between ENP and PCFG whales. We use ENP gray whale length-at-age data comprised of strandings, whaling, and aerial photogrammetry (1926-1997) for comparison to data from PCFG whales collected through non-invasive techniques (2016-2022) to estimate age (photo identification) and length (drone-based photogrammetry). We use Bayesian methods to incorporate uncertainty associated with morphological measurements (manual and photogrammetric) and age estimates. We find that while PCFG and ENP whales have similar growth rates, PCFG whales reach smaller asymptotic lengths. Additionally, PCFG whales have relatively smaller skulls and flukes than ENP whales. These findings represent a striking example of morphological adaptation that may facilitate PCFG whales accessing a foraging niche distinct from the Arctic foraging grounds of the broader ENP population.

Somatic Growth and Maturity for Four Species of River Cooter Including Pseudemys concinna suwanniensis, P. nelsoni, P. peninsularis, and P. texana.

Pseudemys is a genus of commonly occurring freshwater turtles with limited growth information across their long lifespans. We used 11,361 mark-recapture events to estimate the somatic growth rates of P. nelsoni, P. peninsularis, P. concinna suwanniensis, and P. texana from freshwater springs and developed a Bayesian growth model to estimate the species-specific, site-specific, and individual effects on growth. We corroborated evidence for fast juvenile growth and slower adult growth in Pseudemys but found uncommonly fast growth rates, with turtles doubling or tripling in size in the first year. P. texana males had the smallest average maximum size (L∞, 243 mm), while P. c. suwanniensis females had the largest (423 mm). Environmental conditions at springs had significant effects on k, the growth coefficient, but not L∞. We derived, using a ratio of length at maturity to L∞ (71.7% and 87%, males and females), that females matured 1.15-1.57 times older than males except for P. c. suwanniensis, which matured three times older. Given the local abundance declines in many Pseudemys from anthropogenic impacts, this study provides important baseline life history information for Pseudemys species for use in ongoing conservation efforts and presents a novel hierarchical modeling approach using a long-term mark-recapture dataset.

Age, growth and maturity of the yellow stingray (Urobatis jamaicensis), a biannually reproductive tropical batoid.

Urobatis jamaicensis is a coastal batoid species affected by habitat loss and small-scale exploitation from fisheries and the aquarium trade, yet the life-history information available is limited. This is the first study to assess the vertebral centra from 195 stingrays to estimate age and growth patterns, and compare them with the biannual reproductive pattern previously reported for this species. Age-at-size data were compared using five different growth models and found a two-parameter von Bertalanffy growth function (VBGF), the Gompertz model and a modified VBGF fit best for males, females and sexes combined, respectively. Maturity was achieved before 1 year. However, growth did not cease with the onset of maturity, but instead slowed down. Results from marginal increment analysis and edge analysis indicated a nonannual somatic growth pattern with influences from the biannual reproduction cycle where peaks in resource allocation may be focused on ovulation rather than growth during March when larger brood sizes are present, while resources may be allocated more towards growth during August and September when brood sizes are generally smaller. These results may be used as a proxy for species with similar reproductive patterns or for those that lack annual or seasonal growth patterns.

Lost reproductive value reveals a high burden of juvenile road mortality in a long-lived species.

Adult mortality is often the most sensitive vital rate affecting at-risk wildlife populations. Therefore, road ecology studies often focus on adult mortality despite the possibility for roads to be hazardous to juvenile individuals during natal dispersal. Failure to quantify concurrent variation in mortality risk and population sensitivity across demographic states can mislead the efforts to understand and mitigate the effects of population threats. To compare relative population impacts from road mortality among demographic classes, we weighted mortality observations by applying reproductive value analysis to quantify expected stage-specific contributions to population growth. We demonstrate this approach for snapping turtles (Chelydra serpentina) observed on roads at two focal sites in Ontario, Canada, where we collected data for both live and dead individuals observed on roads. We estimated reproductive values using stage-classified matrix models to compare relative population-level impacts of adult and juvenile mortality. Reproductive value analysis is a tractable approach to assessing demographically variable effects for applications covering large spatial scales, nondiscrete populations, or where abundance data are lacking. For one site with long-term life-history data, we compared demographic frequency on roads to expected general population frequencies predicted by the matrix model. Our application of reproductive value is sex specific but, as juvenile snapping turtles lack external secondary sex characters, we estimated the sex ratio of road-crossing juveniles after dissecting and sexing carcasses collected on roads at five sites across central Ontario, Canada. Juveniles were more abundant on roads than expected, suggesting a substantial dispersal contribution, and the road-killed juvenile sex ratio approached 1:1. A higher proportion of juveniles were also found dead compared with adults, and cumulative juvenile mortality had similar population-level importance as adult mortality. This suggests that the impact of roads needs to be considered across all life stages, even in wildlife species with slow life histories, such as snapping turtles, that are particularly sensitive to adult mortality.

Accounting for temporal and individual variation in the estimation of Von Bertalanffy growth curves.

Growth and growth limitation are important indicators of density dependence and environmental limitation of populations. Estimating individual growth trajectories is therefore an important aspect of understanding and predicting the life history and dynamics of a population. Variation in individual growth trajectories arises due to variation in the environmental factors limiting individual growth. This environmental limitation can vary over time, between cohorts and between individuals within a cohort. For a complete and accurate understanding of individual growth in a population, it is important to include all these sources of variation. So far, statistical models only accounted for a subset of these factors or required an extensive growth history of individuals. Here, we present a novel model describing the growth curves of cohorts in a population. This model is derived from a stochastic form of the Von Bertalanffy growth equation describing individual growth. The model is specifically tailored for use on length-at-age data in which the growth trajectory of an individual is unknown and every individual is only measured once. The presented method can also be used if growth limitation differs strongly between age or length classes. We demonstrate the use of the model for length-at-age data of North Sea plaice (Pleuronectes platessa) from the last 30 years. Fitting this model to length-at-age data can provide new insights in the dynamics of the environmental factors limiting individual growth and provides a useful tool for ecological research and management.

Population dynamics of Acentrogobius viridipunctatus (Actinopteri: Gobiidae) in the Vietnamese Mekong Delta.

Spotted green goby Acentrogobius viridipunctatus is one of the commercial gobies living from brackish to freshwater globally, including the Southwest of Viet Nam, but there is no data on population biological traits used for assessing status of stock. Consequently, the research aimed to provide life history parameters of this fish like asymptotic length (L ∞ ), growth coefficient indicator (K), exploitation rate (E), length at first capture (L 50 ), the maximum age (t max ), growth coefficient (Φ'), and mortalities. A total 960 individuals (528 males and 432 females) were caught using trawls at two ecoregions, including Tra Vinh-Soc Trang (TVST) and Bac Lieu-Ca Mau (BLCM), from January to December 2020. Results showed that average fish length was 8.0-11.0 cm, and L ∞ was 14.70 cm at TVST and BLCM. The von Bertalanffy growth curve was L t = 14.70 (1 - e-0.73(t+0.27)) at TVST and L t = 14.70 (1 - e-0.84 (t+0.23)) at BLCM, respectively. The Φ' at BLCM (2.26) was higher than that at TVST (2.20), but t max at BLCM (3.57 yrs) was lower than that at TVST (4.11 yrs). The total mortality (Z), fishing mortality (F), and natural mortality (M) at TVST were 2.09 yr-1, 0.29 yr-1, and 1.80 yr-1, respectively. These parameters at BLCM were larger than those at TVST, with values of 3.47 yr-1, 0.34 yr-1, and 3.14 yr-1, respectively. The fishing status at TVST and BLCM was not subjected to overfishing as E in these regions (0.14 and 0.10) were lower than E 0.1 (0.76 and 0.72). Local authorities should ask fishers to increase fishing mesh size for sustainable exploitation.

Parameter estimation of the Solow-Swan fundamental differential equation.

Background: The Solow-Swan model describes the long-term growth of the capital to labor ratio by the fundamental differential equation of Solow-Swan theory. In conventional approaches, this equation was fitted to data using additional information, such as the rates of population growth, capital depreciation, or saving. However, this was not the best possible fit. Objectives: Using the method of least squares, what is the best possible fit of the fundamental equation to the time-series of the capital to labor ratios? Are the best-fit parameters economically sound? Method: For the data, we used the Penn-World Table in its 2021 version and compared six countries and three definitions of the capital to labor ratio. For optimization, we used a custom-made variant of the method of simulated annealing. We also compared different optimization methods and calibrations. Results: When comparing different methods of optimization, our custom-made tool provided reliable parameter estimates. In terms of R-squared they improved upon the parameter estimates of the conventional approach. Except for the USA, the best-fit values of the exponent were unplausible, as they suggested a too large elasticity of output. However, using a different calibration resulted in more plausible values of the best-fit exponent also for France and Pakistan, but not for Argentina and Japan. Conclusion: Our results have shown a discrepancy between the best-fit parameters obtained from optimization and the parameter values that are deemed plausible in economy. We propose a research program to resolve this issue by investigating if suitable calibrations may generate economically plausible best-fit parameter values.

Orphaning stunts growth in wild African elephants.

Orphans of several species suffer social and physiological consequences such as receiving more aggression from conspecifics and lower survival. One physiological consequence of orphaning, stunted growth, has been identified in both humans and chimpanzees, but has not been assessed in a non-primate species. Here, we tested whether wild African elephant orphans show evidence of stunted growth. We measured individually known female elephants in the Samburu and Buffalo Springs National Reserves of Kenya, with a rangefinder capable of calculating height, to estimate a von Bertalanffy growth curve for female elephants of the study population. We then compared measurements of known orphans and non-orphans of various ages, using a Bayesian analysis to assess variation around the derived growth curve. We found that orphans are shorter for their age than non-orphans. However, results suggest orphans may partially compensate for stunting through later growth, as orphans who had spent a longer time without their mother had heights more similar to non-orphans. More age mates in an individual's family were associated with taller height, suggesting social support from peers may contribute to increased growth. Conversely, more adult females in an individual's family were associated with shorter height, suggesting within-group competition for resources with older individuals may reduce juvenile growth. Finally, we found a counterintuitive result that less rainfall in the first 6 years of life was correlated with taller height, potentially reflecting the unavoidable bias of measuring individuals who were fit enough to survive conditions of low rainfall as young calves. Reduced growth of individuals has been shown to reduce survival and reproduction in other species. As such, stunting in wildlife orphans may negatively affect fitness and represents an indirect effect of ivory poaching on African elephants.

Density-dependence and environmental variability have stage-specific influences on European grayling growth.

Fish somatic growth is indeterminate and can be influenced by a range of abiotic and biotic variables. With climate change forecast to increase the frequency of warming and unusual discharge events, it is thus important to understand how these variables currently influence somatic growth and how that might differ for specific age-classes and/ or life stages. Here, we used a 17-year dataset from a chalk stream in southern England to identify the abiotic and biotic influences on the growth of juvenile, sub-adult and adult life stages of European grayling (Thymallus thymallus), a cold-water riverine salmonid. The results revealed that interannual variations in grayling growth were well described by annual- and site-specific abiotic and biotic explanatory variables. We found divergent responses between life stages to increased temperature and unusual discharge during the main growth period with, for example, elevated temperatures related to increased juvenile growth but reduced sub-adult growth, and high discharge events related to increased sub-adult growth yet reduced juvenile growth. Conversely, stage-specific grayling abundance negatively influenced growth at each life stage, though only juvenile growth was impacted by the abundance of a competitor species, brown trout (Salmo trutta). These results emphasise the merits of testing a wide range of environmental and biological explanatory variables on fish growth, and across life stages. They also reveal the importance of maintaining high habitat heterogeneity in rivers to ensure all life stages can reduce their competitive interactions and have access to adequate flow and thermal refugia during periods of elevated environmental stress.

Conditional Ulam stability and its application to von Bertalanffy growth model.

The purpose of this paper is to apply conditional Ulam stability, developed by Popa, Rașa, and Viorel in 2018, to the von Bertalanffy growth model $ \frac{dw}{dt} = aw^{\frac{2}{3}}-bw $, where $ w $ denotes mass and $ a > 0 $ and $ b > 0 $ are the coefficients of anabolism and catabolism, respectively. This study finds an Ulam constant and suggests that the constant is biologically meaningful. To explain the results, numerical simulations are performed.

How temperature variation affects white-rot fungi mycelial growth dynamics: a nonlinear mixed models approach.

This study aimed to model mycelial growth in a factorial experiment with two species of white rot fungi growing at five temperatures. The nonlinear models evaluated were logistic, Gompertz, Weibull and von Bertalanffy. The adjustments were performed first by evaluating the fixed and mixed-effects models with random effects, added to the fixed parameters. Then, the best adjusted model was improved by an adequated covariance structure, and dummy variables were added to the parameters asymptote (α) and abscissa of the inflection point (ß) in the model in order to verify the effect of the experiment factors, species and temperatures on the regression parameters. The criteria used to compare models were residual variance and Akaike information criterion. Gompertz and von Bertalanffy mixed-effects models were better adjusted. The parameters differed between species. Moreover, the linear and quadratic effects of temperature evaluated in each species were significant. The models were reparametrized in order to consider two parameters of interest: velocity (µ) at the inflection point and lag time (λ). The maximum growth velocity was obtained at 25.4 °C by S. ostrea, while T. villosa was achieved at 30.5 °C; both fungi suffered less lag time by increasing the temperature.

Age frequency, growth, mortality, and PAH levels of roughtongue bass (Pronotogrammus martinicensis) following the Deepwater Horizon oil spill.

Age, growth, mortality, and polycyclic aromatic hydrocarbons (PAHs) in roughtongue bass (Pronotogrammus martinicensis) were examined in the northern Gulf of Mexico following the 2010 Deepwater Horizon oil spill. Fish (N = 1090) were collected from September 2014 to July 2015 from the Alabama Alps (54 km from the spill site) and Roughtongue Reef (111 km from the spill site). Sites were dominated by the 2010 year-class. Growth rates were significantly lower for fish from Alabama Alps compared to Roughtongue Reef (p < 0.001) and likely linked to proximity of the Mississippi River discharge. Mean total PAH ± SD was 50 ± 52.6 and ranged from 0 to 220 ppb. These PAH levels were below a 300-ppb minimum effect level and not significantly different between sites. The dominant 2010 year-class, low PAH levels, and similar growth rates to pre-spill measures indicated that the Deepwater Horizon oil spill had little effect on roughtongue bass.

Evaluation of non-linear models for growth curve in Brazilian tropical goats.

In the present study, 1060 body weight-age records were measured on 300 Canindé goats. The records were collected from Rio Grande do Norte state, North-eastern part of Brazil, and used for evaluation of non-linear models (Brody, Von Bertalanffy, Logistic, and Gompertz) describing growth curve in goats. The non-linear models were compared applying mean squared error of prediction (MSEP), mean squared residue (MSR), mean absolute deviation (MAD), adjusted coefficient of determination (R2aj), and convergence percentage (%C) for determining the most appropriate model describing the growth curve in Canindé goats. The absolute growth rate (AGR) was estimated based on the adjusted model as a function of time to describe the growth rate of the animals, and the effect of sex on the curve parameters was assessed. The Brody model was the most indicated to represent the average growth curve of Canindé goats. Furthermore, the growth curves estimated by the models under study show that the Brody model displayed the best fit, indicating a greater association between observed and estimated weights, suggesting that it can be used to describe the growth behavior and development of the animals analyzed of both sexes.

What is the status of metabolic theory one century after Pütter invented the von Bertalanffy growth curve?

Metabolic theory aims to tackle ecological and evolutionary problems by explicitly including physical principles of energy and mass exchange, thereby increasing generality and deductive power. Individual growth models (IGMs) are the fundamental basis of metabolic theory because they represent the organisational level at which energy and mass exchange processes are most tightly integrated and from which scaling patterns emerge. Unfortunately, IGMs remain a topic of great confusion and controversy about the origins of the ideas, their domain and breadth of application, their logical consistency and whether they can sufficiently capture reality. It is now 100 years since the first theoretical model of individual growth was put forward by Pütter. His insights were deep, but his model ended up being attributed to von Bertalanffy and his ideas largely forgotten. Here I review Pütter's ideas and trace their influence on existing theoretical models for growth and other aspects of metabolism, including those of von Bertalanffy, the Dynamic Energy Budget (DEB) theory, the Gill-Oxygen Limitation Theory (GOLT) and the Ontogenetic Growth Model (OGM). I show that the von Bertalanffy and GOLT models are minor modifications of Pütter's original model. I then synthesise, compare and critique the ideas of the two most-developed theories, DEB theory and the OGM, in relation to Pütter's original ideas. I formulate the Pütter, DEB and OGM models in the same structure and with the same notation to illustrate the major similarities and differences among them. I trace the confusion and controversy regarding these theories to the notions of anabolism, catabolism, assimilation and maintenance, the connections to respiration rate, and the number of parameters and state variables their models require. The OGM model has significant inconsistencies that stem from the interpretation of growth as the difference between anabolism and maintenance, and these issues seriously challenge its ability to incorporate development, reproduction and assimilation. The DEB theory is a direct extension of Pütter's ideas but with growth being the difference between assimilation and maintenance rather than anabolism and catabolism. The DEB theory makes the dynamics of Pütter's 'nutritive material' explicit as well as extending the scheme to include reproduction and development. I discuss how these three major theories for individual growth have been used to explain 'macrometabolic' patterns including the scaling of respiration, the temperature-size rule (first modelled by Pütter), and the connection to life history. Future research on the connections between theory and data in these macrometabolic topics have the greatest potential to advance the status of metabolic theory and its value for pure and applied problems in ecology and evolution.

Biological and statistical interpretation of size-at-age, mixed-effects models of growth.

The differences in life-history traits and processes between organisms living in the same or different populations contribute to their ecological and evolutionary dynamics. We developed mixed-effect model formulations of the popular size-at-age von Bertalanffy and Gompertz growth functions to estimate individual and group variation in body growth, using as a model system four freshwater fish populations, where tagged individuals were sampled for more than 10 years. We used the software Template Model Builder to estimate the parameters of the mixed-effect growth models. Tests on data that were not used to estimate model parameters showed good predictions of individual growth trajectories using the mixed-effects models and starting from one single observation of body size early in life; the best models had R 2 > 0.80 over more than 500 predictions. Estimates of asymptotic size from the Gompertz and von Bertalanffy models were not significantly correlated, but their predictions of size-at-age of individuals were strongly correlated (r > 0.99), which suggests that choosing between the best models of the two growth functions would have negligible effects on the predictions of size-at-age of individuals. Model results pointed to size ranks that are largely maintained throughout the lifetime of individuals in all populations.