A diffusion process to model generalized von Bertalanffy growth patterns: Fitting to real data

The von Bertalanffy growth curve has been commonly used for modeling animal growth (particularly fish). Both deterministic and stochastic models exist in association with this curve, the latter allowing for the inclusion of fluctuations or disturbances that might exist in the system under consideration which are not always quantifiable or may even be unknown. This curve is mainly used for modeling the length variable whereas a generalized version, including a new parameter b≥1, allows for modeling both length and weight for some animal species in both isometric (b=3) and allometric (b≠3) situations.In this paper a stochastic model related to the generalized von Bertalanffy growth curve is proposed. This model allows to investigate the time evolution of growth variables associated both with individual behaviors and mean population behavior. Also, with the purpose of fitting the above-mentioned model to real data and so be able to forecast and analyze particular characteristics, we study the maximum likelihood estimation of the parameters of the model. In addition, and regarding the numerical problems posed by solving the likelihood equations, a strategy is developed for obtaining initial solutions for the usual numerical procedures. Such strategy is validated by means of simulated examples. Finally, an application to real data of mean weight of swordfish is presented.     

Biphasic growth in fish II: empirical assessment

In [Quince, et al., 2008. Biphasic growth in fish I: Theoretical foundations. J. Theor. Biol., doi:10.1016/j.jtbi.2008.05.029], we developed a set of biphasic somatic growth models, where maturation is accompanied by a deceleration of growth due to allocation of energy to reproduction. Here, we use growth data from both hatchery-raised and wild populations of a large freshwater fish (lake trout, Salvelinus namaycush) to test these models. We show that a generic biphasic model provides a better fit to these data than the von Bertalanffy model. We show that the observed deceleration of somatic growth in females varies directly with gonad weight at spawning, with observed egg volumes roughly 50% of the egg volumes predicted under the unrealistic assumption of perfectly efficient energy transfer from somatic lipids to egg lipids. We develop a Bayesian procedure to jointly fit a biphasic model to observed growth and maturity data. We show that two variants of the generic biphasic model, both of which assume that annual allocation to reproduction is adjusted to maximise lifetime reproductive output, provide complementary fits to wild population data: maturation time and early adult growth are best described by a model with no constraints on annual reproductive investment, while the growth of older fish is best described by a model that is constrained so that the ratio of gonad size to somatic weight (g) is fixed. This behaviour is consistent with the additional observation that g increases with size and age among younger, smaller breeding females but reaches a plateau among older, larger females. We then fit both of these optimal models to growth and maturation data from nineteen wild populations to generate population-specific estimates of ‘adapted mortality’ rate: the adult mortality consistent with observed growth and maturation schedules, given that both schedules are adapted to maximise lifetime reproductive output. We show that these estimates are strongly correlated with independent estimates of the adult mortality experienced by these populations.     

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.     

The growth of the freshwater murrel,Ophicephalus punctatus Bloch

Summary The growth of O. punctatus has been studied from the zones on the opercular bones and scales. Size estimates obtained from the opercular bones and scales were further substantiated by length frequency distribution and back-calculations. A close agreement in length-age relationship was obtained by various methods. These observations provide adequate evidence towards the validity of age determination in O. punctatus.Growth rate differs markedly in the two sexes. Males grow faster than the females. To study the changes in length with age, von Bertalanffy growth equation and Gompertz curve were used. The theoretical lengths obtained from the von Bertalanffy equation agreed very closely with the observed lengths.There is sexual difference in the weight-length relationship of O. punctatus. Modal weight of each year class obtained according to age reading showed that growth in weight is faster in males. The theoretical growth equation gives a good fit for weight-age data. O. punctatus is generally a fish of the impounded waters. The interrelationship between pond environment and growth characteristics has been discussed.     

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.     

Age and growth of longnose trevally (Carangoides chrysophrys) in the Arabian Sea

The ages and growth of longnose trevally (Carangoides chrysophrys), caught in the northwest Arabian Sea between April 2005 and September 2006, were investigated. Age and growth of 336 fish specimens were determined using sectioned sagittae otoliths. Annual opaque growth rings were formed between December and March, with the majority being laid down in February and March, coinciding with the spawning period and high water temperatures. Marginal zone or edge analysis was used to validate the annual deposition of the opaque zone in the otoliths. This species showed large variations in length‐at‐age, suggesting large growth variations among individuals of the same cohort. The estimated von Bertalanffy growth model differed significantly between the sexes, with males having larger mean lengths at age and reaching a larger asymptotic size. The von Bertalanffy growth models were TL (cm) = 73.34[1‐exp (?0.25 (t + 1.21))] and TL(cm) = 73.26[1‐exp (?0.24 (t + 1.20))] for males and females, respectively.     

Modelling the growth of Japanese eel Anguilla japonica in the lower reach of the Kao-Ping River, southern Taiwan: an information theory approach

Information theory was applied to select the best model fitting total length ( L _T)-at-age data and calculate the averaged model for Japanese eel Anguilla japonica compiled from published literature and the differences in growth between sexes were examined. Five candidate growth models were the von Bertalanffy, generalized von Bertalanffy, Gompertz, logistic and power models. The von Bertalanffy growth model with sex-specific coefficients was best supported by the data and nearly overlapped the averaged growth model based on Akaike weights, indicating a similar fit to the data. The Gompertz, generalized von Bertalanffy and power growth models were also substantially supported by the data. The L _T at age of A. japonica were larger in females than in males according to the averaged growth mode, suggesting a sexual dimorphism in growth. Model inferences based on information theory, which deal with uncertainty in model selection and robust parameter estimates, are recommended for modelling the growth of A. japonica .     

Stochastic von Bertalanffy models, with applications to fish recruitment

We consider three individual-based models describing growth in stochastic environments. Stochastic differential equations (SDEs) with identical von Bertalanffy deterministic parts are formulated, with a stochastic term which decreases, remains constant, or increases with organism size, respectively. Probability density functions for hitting times are evaluated in the context of fish growth and mortality. Solving the hitting time problem analytically or numerically shows that stochasticity can have a large positive impact on fish recruitment probability. It is also demonstrated that the observed mean growth rate of surviving individuals always exceeds the mean population growth rate, which itself exceeds the growth rate of the equivalent deterministic model. The consequences of these results in more general biological situations are discussed.     

Age and growth estimates for the smooth skate, Malacoraja senta, in the Gulf of Maine

Age and growth estimates for the smooth skate, Malacoraja senta, were derived from 306 vertebral centra from skates caught in the North Atlantic off the coast of New Hampshire and Massachusetts, USA. Males and females were aged to 15 and 14 years, respectively. Male and female growth diverged at both ends of the data range and the sexes required different growth functions to describe them. Males followed a traditional growth scenario and were best described by a von Bertalanffy curve with a set L _o (11 cm TL) where L _inf  = 75.4 cm TL, K = 0.12. Females required the use of back-calculated values to account for a lack of small individuals, using these data they were best described by a von Bertalanffy curve where growth parameters derived from vertebral length-at-age data are L _inf  = 69.6 cm TL, K = 0.12, and L _o  = 10.     

The importance of juveniles in modelling growth: butterflyfish at Lizard Island

Synopsis I established and fitted von Bertalanffy growth functions to size-at-age data for four species of chaetodontids at Lizard Island. Special emphasis on juveniles provided detailed information of the early growth period. All four species demonstrated rapid initial growth achieving an average of 92% of maximum theoretical size in the first 2years. I used various constraints of the theoretical age at length zero (t₀) in an analysis of both complete data sets and data sets using only adult fish. An unconstrained value of t₀ resulted in the best-fit (maximum r²) curve when juveniles were included. When excluding juveniles, it was necessary to constrain t₀ to an approximate settling size to most closely represent the growth of the species.     

Time in tortoiseshell: a bomb radiocarbon-validated chronology in sea turtle scutes

Some of the most basic questions of sea turtle life history are also the most elusive. Many uncertainties surround lifespan, growth rates, maturity and spatial structure, yet these are critical factors in assessing population status. Here we examine the keratinized hard tissues of the hawksbill (Eretmochelys imbricata) carapace and use bomb radiocarbon dating to estimate growth and maturity. Scutes have an established dietary record, yet the large keratin deposits of hawksbills evoke a reliable chronology. We sectioned, polished and imaged posterior marginal scutes from 36 individual hawksbills representing all life stages, several Pacific populations and spanning eight decades. We counted the apparent growth lines, microsampled along growth contours and calibrated Δ¹⁴C values to reference coral series. We fit von Bertalanffy growth function (VBGF) models to the results, producing a range of age estimates for each turtle. We find Hawaii hawksbills deposit eight growth lines annually (range 5–14), with model ensembles producing a somatic growth parameter (k) of 0.13 (range 0.1–0.2) and first breeding at 29 years (range 23–36). Recent bomb radiocarbon values also suggest declining trophic status. Together, our results may reflect long-term changes in the benthic community structure of Hawaii reefs, and possibly shed light on the critical population status for Hawaii hawksbills.     

Population dynamics,growth and reproductive rates of Littorina nigrolineata Gray from a moderately sheltered locality in North Wales

The population dynamics of Littorina nigrolineata Gray on a moderately sheltered boulder shore was characterized by an almost constant mortality rate of post-juvenile snails both within and between years. Cohorts of post-juvenile snails had a “half life” of about 5 months. Juvenile snails had higher mortality rates than post-juveniles. Mortality rate was independent of size in post-juveniles. The population of post-juvenile snails fluctuated only about two-fold during the 3 yr of study. Maxima of population size occurred each September as a result of an autumnal pulse of recruitment into the post-juvenile size classes. These recruits originated from eggs laid in the summer of the previous year. The size-frequency structure of the population lacked well defined modes except during the spring and summer when the pulse of recruits formed a distinct mode, which later disappeared due to growth and mortality of the recruits. Oviposition continued throughout the year with a pulse in summer. Newly laid eggs took 4–7 wk to hatch. The hatchlings had a mean shell height of 0.5 mm and were estimated to take 6–9 months to grow to a shell height of about 4 mm. Growth increments over periods of 3 months. estimated from tagged cohorts of snails ranging from 3 to 16 mm, released on a stone jetty, revealed seasonal and size-specific changes in growth rate. Growth rate was significantly lower between December and March than at other times of the year. Seasonal differences in growth rate did not, however, significantly affect the yearly size increments of snails hatching in summer or winter. Growth increments did not decline linearly with increasing body size, as is assumed by the von Bertalanffy growth model, but increased to a maximum at about 6–7 mm shell height. Beyond the maximum, the growth increments decreased linearly with increasing shell height. This non-linearity caused an inflexion in the growth curve, so that the shell height. This non-linearity caused an inflexion in the and 12.4 mm by the von Bertalanffy growth curve. The deviation from the von Bertalanffy growth model was possibly due to departures from the assumptions that the assimilation rate is proportional to the square of a linear size dimension and that the metabolic rate is proportional to the cube of the same dimension. A von Bertalanffy growth curve, fitted to growth increments measured from “growth checks” on tagged snails on the boulder shore, predicted slightly faster growth rates than the von Bertalanffy growth curve fitted to increments of mean cohort size on the stone jetty. The closeness of the two curves, however, suggested that both methods of measuring growth increments were tolerably accurate.     

Daily age determination and growth rates of freshwater fish throughout a regulated lotic system of the Murray‐Darling Basin Australia

To facilitate future research in freshwater fish recruitment response to environmental flow delivery, size‐at‐age and growth models are presented for eight fish species occurring in south‐eastern Australia; three small‐bodied species and the juvenile 0+ age classes of five large‐bodied species. Otolith increments were used to estimate the daily age of golden perch Macquaria ambigua, bony bream Nematalosa erebi, common carp Cyprinus carpio; Murray cod Maccullochella peelii, freshwater eel‐tailed catfish Tandanus tandanus, Australian smelt Retropinna semoni, un‐specked hardyhead Craterocephalus stercusmuscarum fulvus and Murray–Darling rainbowfish Melanotaenia fluviatilis. Linear growth models provided the best fit for length‐at‐age data of juvenile 0+ age large‐bodied species; whereas von Bertalanffy growth functions provided the best fit to length‐at‐age data of small‐bodied species. The results provide novel baseline data for future research in this area.     

Age, growth, maturity, and mortality of the Alaska skate, Bathyraja parmifera, in the eastern Bering Sea

The Alaska skate, Bathyraja parmifera, is the most abundant species of skate on the eastern Bering Sea shelf, accounting for over 90% of total skate biomass. However, little is known regarding the life history of this species despite its common occurrence as bycatch in several Bering Sea fisheries. This is the first study to focus on the age and growth of B. parmifera. From 2003 to 2005, more than one thousand specimens were collected by fisheries observers and on scientific groundfish surveys. Annual banding patterns in more than 500 thin sections of vertebral centra were examined for age determination. Caudal thorns were tested as a potentially non-lethal ageing structure. Annual band pair deposition was verified through edge and marginal increment analyses. A three-parameter von Bertalanffy growth function and a Gompertz growth function were fit to observed length-at-age data. Both models provided significant fits, although the Gompertz function best described the overall pattern of growth in both males and females, based upon statistical criteria and parameter estimates. Age and size at 50% maturity were 9 years and 92 cm TL for males and 10 years and 93 cm TL for females. The maximum observed ages for males and females were 15 years and 17 years, respectively. Estimates of natural mortality (M) ranged from 0.14 to 0.28, and were based on published relationships between M and longevity, age at maturity, and the von Bertalanffy growth coefficient. Due to these life history characteristics and a lack of long-term species-specific stock data, a conservative management approach would be appropriate for B. parmifera.     

Growth characteristics of freshwater pearl mussels,Margaritifera margaritifera (L.)

• 1 Shell growth in the freshwater pearl mussel, Margaritifera margaritifera, was investigated. Three non‐linear growth models (i.e. power, logistic and von Bertalanffy) were fitted to Scottish length‐at‐age data sets and compared.
• 2 Overall, the von Bertalanffy model outperformed the other two approaches, generating the smallest residuals in eight out of 11 samples (the logistic model provided slightly better fits to the other three). It was concluded that individual M. margaritifera appear to grow in an approximately asymptotic fashion and that the von Bertalanffy equation is an appropriate growth model to fit to freshwater pearl mussel length‐at‐age data.
• 3 The ranges in von Bertalanffy parameter estimates observed (k = 0.023–0.075 year^‐1, L_∞ = 77–158 mm, t_o = ‐3.93–4.33 years) are typical of those reported in northern European populations.
• 4 Most of the populations investigated had relatively low k‐values and high maximum age (A_max) estimates. This feature, which suggests high long‐term productivity and less vulnerability to decline (i.e. larger, longer‐living mussels produce more offspring), may be a reason why these populations have survived until now. The population which appears to be the most vulnerable (i.e. which has the highest k and lowest A_max) is probably not recruiting adequately at present.
• 5 An index of absolute growth (mean shell length‐at‐age) was also used for comparing different populations. Observed between‐ and within‐river differences in mussel growth patterns may be associated with a number of environmental factors, particularly water temperature and productivity.
• 6 A significant positive relationship between river length and mean mussel length‐at‐age was observed. In general, mussels grow large in large, cold rivers and vice versa, although there are exceptions which suggest that additional factors may be involved.