Estimating von Bertalanffy parameters with individual and environmental variations in growth

Urbanization effects on growth and otolith asymmetry in Chrysichthys nigrodigitatus and Oreochromis niloticus within tropical coastal lagoon watersheds

In this study, we investigated the growth dynamics and otolith shape asymmetry of two fish species, Chrysichthys nigrodigitatus (CN) and Oreochromis niloticus (ON), within urbanized watersheds of the southern lagoon system, Nigeria. Using the von Bertalanffy growth model (VBGM), in addition to sediment metal concentration indices such as the average shale content, index of geoaccumulation (Igeo), contamination factor (CF), pollution load index (PLI), and potential ecological risk (PER) index, contamination levels were classified, and ecological risks were assessed. Notably, a lower growth potential (t0) was observed in CN at Ikorodu than at Epe, with similar trends for ON in the Epe during the dry season. Otolith asymmetry patterns, particularly in the CN at Ikorodu and ON in the Epe during the dry season, exhibited distinct ecological variations, indicating heightened stress levels at Ikorodu. Sediment analyses revealed moderate to strong contamination (Cd, Pb, Ni, and Cr) in both Lagos Lagoon (Ikorodu) and Epe Lagoon, with Ikorodu exhibiting notably high to moderate contamination levels according to the CF index. Elevated PLI values for Cd and Pb in Ikorodu, in addition to greater PER, indicated increased risk, with Cd posing a high risk (61.42%) and Pb posing a moderate risk (49.50%). Additionally, the reduced asymptotic length in the Epe during the dry season suggests that Chrysichthys nigrodigitatus is adaptable to seasonal variations, while divergent growth patterns in both areas indicate the existence of trade-off mechanisms in response to changing conditions. Habitat-specific otolith asymmetry and metal contamination underscore species adaptability, with wider stressor variability in Lagos than in Epe. Furthermore, multidimensional scaling analysis highlights the intricate relationship between otolith shape variables and environmental factors, emphasizing the need for tailored conservation efforts in urbanized watersheds.

Predicting age and mass at maturity from feeding behavior and diet in Manduca sexta: An empirical test of a life history model

Feeding for most animals involves bouts of active ingestion alternating with bouts of no ingestion. In insects, the temporal patterning of bouts varies widely with resource quality and is known to affect growth, development time, and fitness. However, the precise impacts of resource quality and feeding behavior on insect life history traits are poorly understood. To explore and better understand the connections between feeding behavior, resource quality, and insect life history traits, we combined laboratory experiments with a recently proposed mechanistic model of insect growth and development for a larval herbivore, Manduca sexta. We ran feeding trials for 4th and 5th instar larvae across different diet types (two hostplants and artificial diet) and used these data to parameterize a joint model of age and mass at maturity that incorporates both insect feeding behavior and hormonal activity. We found that the estimated durations of both feeding and nonfeeding bouts were significantly shorter on low-quality than on high-quality diets. We then explored how well the fitted model predicted historical out-of-sample data on age and mass of M. sexta. We found that the model accurately described qualitative outcomes for the out-of-sample data, notably that a low-quality diet results in reduced mass and later age at maturity compared with high-quality diets. Our results clearly demonstrate the importance of diet quality on multiple components of insect feeding behavior (feeding and nonfeeding) and partially validate a joint model of insect life history. We discuss the implications of these findings with respect to insect herbivory and discuss ways in which our model could be improved or extended to other systems.

Juvenile-to-adult transition invariances in fishes: Perspectives on proximate and ultimate causation

To bridge physiological and evolutionary perspectives on size at maturity in fishes, the authors focus on the approximately invariant ratio between the estimated oxygen supply at size at maturity (Q_m ) relative to that at asymptotic size (Q_∞ ) among species within a taxonomic group, and show how two important theories related to this phenomenon complement each other. Gill-oxygen limitation theory proposes a mechanistic basis for a universal oxygen supply-based threshold for maturation, which applies among and within species. On the contrary, the authors show that a generalisation of life-history theory for the invariance of size at maturity (L_m ) relative to asymptotic size (L_∞ ) can provide an evolutionary rationale for an oxygen-limited maturation threshold (Q_m /Q_∞ ). Extending previous inter- and intraspecific analyses, the authors show that maturation invariances also occur in lake whitefish Coregonus clupeaformis (Mitchill 1818), but at both scales, theory seems to underestimate the value of the maturation threshold. They highlight some key uncertainties in the model that should be addressed to help resolve the mismatch.

Using Observed Residual Error Structure Yields the Best Estimates of Individual Growth Parameters

Obtaining the best possible estimates of individual growth parameters is essential in studies of physiology, fisheries management, and conservation of natural resources since growth is a key component of population dynamics. In the present work, we use data of an endangered fish species to demonstrate the importance of selecting the right data error structure when fitting growth models in multimodel inference. The totoaba (Totoaba macdonaldi) is a fish species endemic to the Gulf of California increasingly studied in recent times due to a perceived threat of extinction. Previous works estimated individual growth using the von Bertalanffy model assuming a constant variance of length-at-age. Here, we reanalyze the same data under five different variance assumptions to fit the von Bertalanffy and Gompertz models. We found consistent significant differences between the constant and nonconstant error structure scenarios and provide an example of the consequences using the growth performance index ϕ′ to show how using the wrong error structure can produce growth parameter values that can lead to biased conclusions. Based on these results, for totoaba and other related species, we recommend using the observed error structure to obtain the individual growth parameters.

Reconciling larval and adult sampling methods to model growth across life-stages

Individual growth rates are intrinsically related to survival and lifetime reproductive success and hence, are key determinants of population growth. Efforts to quantify age-size relationships are hampered by difficulties in aging individuals in wild populations. In addition, species with complex life-histories often show distinct shifts in growth that cannot be readily accommodated by traditional modelling techniques. Amphibians are often characterized by rapid larval growth, cessation of growth prior to metamorphosis, and resumption of growth in the adult stage. Compounding issues of non-linear growth, amphibian monitoring programs typically sample larval and adult populations using dissimilar methods. Here we present the first multistage growth model that combines disparate data collected across life-history stages. We model the growth of the endangered Reticulated Flatwoods Salamander, Ambystoma bishopi, in a Bayesian framework, that accounts for unknown ages, individual heterogeneity, and reconciles dip-net and drift fence sampling designs. Flatwoods salamanders achieve 60% of growth in the first 3 months of life but can survive for up to 13 years as a terrestrial adult. We find evidence for marked variability in growth rate, the timing and age at metamorphosis, and maximum size, within populations. Average size of metamorphs in a given year appeared strongly dependent on hydroperiod, and differed by >10mm across years with successful recruitment. In contrast, variation in the sizes of emerging metamorphs appeared relatively constant across years. An understanding of growth will contribute to the development of population viability analyses for flatwoods salamanders, will guide management actions, and will ultimately aid the recovery of the species. Our model formulation has broad applicability to amphibians, and likely any stage-structured organism in which homogenous data cannot be collected across life-stages. The tendency to ignore stage-structure or omit non-conforming data in growth analyses can no longer be afforded given the high stakes of management decisions, particularly for endangered or at-risk populations.

A Stochastic Model for Predicting Age and Mass at Maturity of Insects

Variation in age and mass at maturity is commonly observed in populations, even among individuals with the same genetic and environmental backgrounds. Accounting for such individual variation with a stochastic model is important for estimating optimal evolutionary strategies and for understanding potential trade-offs among life-history traits. However, most studies employ stochastic models that are either phenomenological or account for variation in only one life-history trait. We propose a model based on the developmental biology of the moth Manduca sexta that accounts for stochasticity in two key life-history traits, age and mass at maturity. The model is mechanistic, describing feeding behavior and common insect developmental processes, including the degradation of juvenile hormone prior to molting. We derive a joint probability density function for the model and explore how the distribution of age and mass at maturity is affected by different parameter values. We find that the joint distribution is generally nonnormal and highly sensitive to parameter values. In addition, our model predicts previously observed effects of temperature change and nutritional quality on the expected values of insect age and mass. Our results highlight the importance of integrating multiple sources of stochasticity into life-history models.

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.

A New Flexible Sigmoidal Growth Model

Biological growth is driven by numerous functions, such as hormones and mineral nutrients, and is also involved in various ecological processes. Therefore, it is necessary to accurately capture the growth trajectory of various species in ecosystems. A new sigmoidal growth (NSG) model is presented here for describing the growth of animals and plants when the assumption is that the growth rate curve is asymmetric. The NSG model was compared with four classic sigmoidal growth models, including the logistic equation, Richards, Gompertz, and ontogenetic growth models. Results indicated that all models fit well with the empirical growth data of 12 species, except the ontogenetic growth model, which only captures the growth of animals. The estimated maximum asymptotic biomass wmax of plants from the ontogenetic growth model was not reliable. The experiment result shows that the NSG model can more precisely estimate the value and time of reaching maximum biomass when growth rate becomes close to zero near the end of growth. The NSG model contains three other parameters besides the value and time of reaching maximum biomass, and thereby, it can be difficult to assign initial values for parameterization using local optimization methods (e.g., using Gauss–Newton or Levenberg–Marquardt methods). We demonstrate the use of a differential evolution algorithm for resolving this issue efficiently. As such, the NSG model can be applied to describing the growth patterns of a variety of species and estimating the value and time of achieving maximum biomass simultaneously.

Estimating thermal performance curves from repeated field observations

Estimating thermal performance of organisms is critical for understanding population distributions and dynamics and predicting responses to climate change. Typically, performance curves are estimated using laboratory studies to isolate temperature effects, but other abiotic and biotic factors influence temperature-performance relationships in nature reducing these models' predictive ability. We present a model for estimating thermal performance curves from repeated field observations that includes environmental and individual variation. We fit the model in a Bayesian framework using MCMC sampling, which allowed for estimation of unobserved latent growth while propagating uncertainty. Fitting the model to simulated data varying in sampling design and parameter values demonstrated that the parameter estimates were accurate, precise, and unbiased. Fitting the model to individual growth data from wild trout revealed high out-of-sample predictive ability relative to laboratory-derived models, which produced more biased predictions for field performance. The field-based estimates of thermal maxima were lower than those based on laboratory studies. Under warming temperature scenarios, field-derived performance models predicted stronger declines in body size than laboratory-derived models, suggesting that laboratory-based models may underestimate climate change effects. The presented model estimates true, realized field performance, avoiding assumptions required for applying laboratory-based models to field performance, which should improve estimates of performance under climate change and advance thermal ecology.

Individual heterogeneity and early life conditions shape growth in a freshwater top predator

Body size can have profound impacts on survival, movement, and reproductive schedules shaping individual fitness, making growth a central process in ecological and evolutionary dynamics. Realized growth is the result of a complex interplay between life history schedules, individual variation, and environmental influences. Integrating all of these aspects into growth models is methodologically difficult, depends on the availability of repeated measurements of identifiable individuals, and consequently represents a major challenge in particular for natural populations. Using a unique 30-yr time series of individual length measurements inferred from scale year rings of wild brown trout, we develop a Bayesian hierarchical model to estimate individual growth trajectories in temporally and spatially varying environments. We reveal a gradual decrease in average juvenile growth, which has carried over to adult life and contributed to decreasing sizes observed at the population level. Commonly studied environmental drivers like temperature and water flow did not explain much of this trend and overall persistent and among-year individual variation dwarfed temporal variation in growth patterns. Our model and results are relevant to a wide range of questions in ecology and evolution requiring a detailed understanding of growth patterns, including conservation and management of many size-structured populations.

Trade-offs between accuracy and interpretability in von Bertalanffy random-effects models of growth

Better understanding of variation in growth will always be an important problem in ecology. Individual variation in growth can arise from a variety of processes; for example, individuals within a population vary in their intrinsic metabolic rates and behavioral traits, which may influence their foraging dynamics and access to resources. However, when adopting a growth model, we face trade-offs between model complexity, biological interpretability of parameters, and goodness of fit. We explore how different formulations of the von Bertalanffy growth function (vBGF) with individual random effects and environmental predictors affect these trade-offs. In the vBGF, the growth of an organism results from a dynamic balance between anabolic and catabolic processes. We start from a formulation of the vBGF that models the anabolic coefficient (q) as a function of the catabolic coefficient (k), a coefficient related to the properties of the environment (γ) and a parameter that determines the relative importance of behavior and environment in determining growth (ψ). We treat the vBGF parameters as a function of individual random effects and environmental variables. We use simulations to show how different functional forms and individual or group variability in the growth function's parameters provide a very flexible description of growth trajectories. We then consider a case study of two fish populations of Salmo marmoratus and Salmo trutta to test the goodness of fit and predictive power of the models, along with the biological interpretability of vBGF's parameters when using different model formulations. The best models, according to AIC, included individual variability in both k and γ and cohort as predictor of growth trajectories, and are consistent with the hypothesis that habitat selection is more important than behavioral and metabolic traits in determining lifetime growth trajectories of the two fish species. Model predictions of individual growth trajectories were largely more accurate than predictions based on mean size-at-age of fish. Our method shares information across individuals, and thus, for both fish populations investigated, allows using a single measurement early in the life of individual fish or cohort to obtain accurate predictions of lifetime individual or cohort size-at-age.

Determining individual variation in growth and its implication for life-history and population processes using the empirical Bayes method

The differences in demographic and life-history processes between organisms living in the same population have important consequences for ecological and evolutionary dynamics. Modern statistical and computational methods allow the investigation of individual and shared (among homogeneous groups) determinants of the observed variation in growth. We use an Empirical Bayes approach to estimate individual and shared variation in somatic growth using a von Bertalanffy growth model with random effects. To illustrate the power and generality of the method, we consider two populations of marble trout Salmo marmoratus living in Slovenian streams, where individually tagged fish have been sampled for more than 15 years. We use year-of-birth cohort, population density during the first year of life, and individual random effects as potential predictors of the von Bertalanffy growth function's parameters k (rate of growth) and (asymptotic size). Our results showed that size ranks were largely maintained throughout marble trout lifetime in both populations. According to the Akaike Information Criterion (AIC), the best models showed different growth patterns for year-of-birth cohorts as well as the existence of substantial individual variation in growth trajectories after accounting for the cohort effect. For both populations, models including density during the first year of life showed that growth tended to decrease with increasing population density early in life. Model validation showed that predictions of individual growth trajectories using the random-effects model were more accurate than predictions based on mean size-at-age of fish.

Somatic growth is a crucial determinant of ecological and evolutionary dynamics, since larger organisms often have higher survival and reproductive success. Size may be the result of intrinsic (i.e. genetic), environmental (temperature, food), and social (competition with conspecifics) factors and interaction between them. Knowing the contribution of intrinsic, environmental, and social factors will improve our understanding of individual population dynamics, help conservation and management of endangered species, and increase our ability to predict future growth trajectories of individuals and populations. The latter goal is also relevant for humans, since predicting future growth of newborns may help identify early pathologies that occur later in life. However, teasing apart the contribution of individual and environmental factors requires powerful and efficient statistical methods, as well as biological insights and the use of longitudinal data. We developed a novel statistical approach to estimate and separate the contribution of intrinsic and environmental factors to lifetime growth trajectories, and generate hypotheses concerning the life-history strategies of organisms. Using two fish populations as a case study, we show that our method predicts future growth of organisms with substantially greater accuracy than using historical information on growth at the population level, and help us identify year-class effects, probably associated with climatic vagaries, as the most important environmental determinant of growth.