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.     

Biological reasons for the unsuitability of growth parameters and indices for comparing fish growth

Parameters k and L in the Von Bertalanffy equation and their derivatives =kL, = logk + 2logL; as well as slopes b_L/t of the regression L=at^b (t, years), relative increments (CI=[L₂–L₁]/L₁), specific growth rates (C_v=InL₂–InL₁), growth characteristics (Clh = CvxL₁) and growth constants (Clt=Cvx[t₂ + t₁]/2) were analyzed. A total of 121 bream Abramis brama stocks in the first 10 years of life were studied. At the same real growth rate (the average absolute linear annual increments, mm year^–1) the values of k, L, , , b_L/t, CI, Cv and Clt in different stocks vary within almost the whole range. The main reason is the natural process of growth self-regulation: the relation between the average body lengths in the first year (L₁) and the relative growth rates (slopes b_L/t) is negative (b_L/t = exp[0.1183–0.0053L₁], r=0.76). The above relation defines 4 principal types of the Ford-Walford lines. Thirty four percent of the stocks have rather steep slopes of the lines and even parallel the absolute slope of 45°, so the L values of these stocks have no biological significance. The authors recommend a simple and more precise, from a biological point of view, approach for comparing fish population growth rates.     

Quantitative analysis of published data on the growth,metabolism, food consumption,and related features of the red-bellied piranha,Serrasalmus nattereri (Characidae)

Synopsis A tentative set of growth parameters of the von Bertalanffy growth equation were estimated for the red-bellied piranha,Serrasalmus nattereri, a common characid of the Amazonas and adjacent floodplains, based on length-frequency data collected by R.H. Lowe-McConnell in Guyana. These parameters and related statistics are then used, along with published data from metabolic, field and feeding experiment data to estimate the relative food consumption of a population ofS. nattereri. This is complemented with biological data assembled from the scattered literature onS. nattereri to provide a snapshot of this species.     

Is the temperature-size rule mediated by oxygen in aquatic ectotherms?

Temperature is an important environmental factor that influences key traits like body size, growth rate and maturity. Ectotherms reared under high temperatures usually show faster growth, but reach a smaller final size, a phenomenon known as the temperature-size rule (TSR). Oxygen may become a limiting resource at high temperatures, when demand for oxygen is high, especially in water as oxygen uptake is far more challenging under water than in air. Therefore, in aquatic ectotherms, the TSR might very well be mediated by temperature effects on oxygen availability and oxygen demand. To distinguish between the direct effects of temperature and oxygen mediated effects, growth rate and final size were measured in the aquatic ectotherm Asellus aquaticus (Linnaeus, 1758) reared under different temperature and oxygen conditions in a factorial design. Growth could be best described by a modified Von Bertalanffy growth function. Both temperature and oxygen affected age at maturity and growth. Growth responses to temperature were dependent on oxygen conditions (interactive effect of temperature and oxygen). Only under hypoxic conditions, when oxygen was most limiting, did we find a classic TSR. Moreover, when comparing treatments differing in temperature, but where the balance between oxygen demand and supply was similar, high temperature increased both growth rate and final size. Thus effects of oxygen may resolve the life-history puzzle of the TSR in aquatic ectotherms.     

A skeletochronological study of age, growth and longevity in a population of the frog Rana ridibunda from southern Europe

Age at sexual maturity and longevity in a population of Rana ridibunda from north-eastern Greece were studied by skeletochronology performed on the phalanges. Analysis of the age structure was based on counting the lines of arrested growth (LAGs). Sexual maturity for both sexes arises during the first year or after the first hibernation. Ages ranged from 1 to 5 years (mean=2.96) among 52 males and from 1 to 5 years (mean=3.73) among 56 females. The mean snout-vent length was 69.03+/-12.6mm in males and 82.38+/-13.27 mm in females. The difference between the sexes in age and size was significant. Growth of individuals was fitted on? The von Bertalanffy model. The growth coefficient (K) was 0.57 in males and 0.54 in females, mainly due to faster male growth between metamorphosis and maturation.     

Age structure, growth and survival rates of the commercial fish Prochilodus nigricans (bocachico) in North-eastern Ecuador

The bocachico, Prochilodus nigricans, is a commercially important fish distributed throughout the central and western Amazon. Age and growth of this species in eastern Ecuador were determined by otolith and scale increment analyses. For otolith analyses, we used thin sections of the astericus, and clear visualization of annuli required the use transmitted polarized light. Accuracy and precision of age estimation were compared to select the best ageing structure. Precision indices showed that otoliths are more reliable than scales to determine the age of bocachicos. Based on marginal increments analysis, we found that annulus formation in both structures occurs once a year between August and December. Seasonal changes in growth were associated with hydrological cycle of the river. The Ecuadorian Prochilodus at the juvenile–adult stage grew faster (i.e., K = 0.29 for ages 2 and older) than conspecific populations from Bolivia and Brazil (K = 0.5). In addition, annual mortality rate (A = 0. 56) was lower than reported for Prochilodus populations elsewhere in the Amazon. We conclude that this population is presently not overexploited, but conservation and management schemes for this population will need to consider that it is an international, trans-boundary migrant.     

Polyfluorochrome marking slows growth only during the marking month in the green sea urchin Strongylocentrotus droebachiensis

Abstract. Used singly, the fluorochrome tags tetracycline and calcein have yielded important insights into sea urchin biology, especially regarding growth. We present a new method of tagging using sequential fluorochrome markers, as well as a more precise method of quantifying growth. Such polyfluorochromes enable repeated markings that allow measurement of multiple growth points and unique identification of individuals or groups. We marked sea urchins, Strongylocentrotus droebachiensis , with four fluorochromes: alizarin complexone, calcein, calcein blue, and tetracycline. All fluorochromes marked both by injection and immersion. We examined the clarity of the mark produced with low, metabolically scaled doses, and higher doses similar to those that have been previously used. We tested the effect of fluorochromes on survival, growth, jaw size, and gonad size by marking a size range (3.9–44.3 mm in diameter) of urchins with either one or all four fluorochromes. We quantified growth using a nominal diameter, that is a fitted constant, times the cube root of weight, which increased the precision of measurements by a factor of six relative to measured diameter. Growth rate was a decreasing function of diameter except for a growth lag in the smallest urchins. Growth rate data for all sizes were fitted using: gamma distributions; Tanaka functions; and, for larger sizes, straight lines (von Bertalanffy model). All treatments produced clear marks, with higher doses producing more reliably clear marks. Tetracycline marking did not affect growth; other treatments produced only transient slowing of growth in the marking month. Growth rate, survival, gonad production, and jaw weight did not differ between control and treatments during the following 6 months. Thus, polyfluorochromes produce reliable marks that do not significantly affect growth or gonad production.     

Individualism in plant populations: using stochastic differential equations to model individual neighbourhood-dependent plant growth

We study individual plant growth and size hierarchy formation in an experimental population of Arabidopsis thaliana, within an integrated analysis that explicitly accounts for size-dependent growth, size- and space-dependent competition, and environmental stochasticity. It is shown that a Gompertz-type stochastic differential equation (SDE) model, involving asymmetric competition kernels and a stochastic term which decreases with the logarithm of plant weight, efficiently describes individual plant growth, competition, and variability in the studied population. The model is evaluated within a Bayesian framework and compared to its deterministic counterpart, and to several simplified stochastic models, using distributional validation. We show that stochasticity is an important determinant of size hierarchy and that SDE models outperform the deterministic model if and only if structural components of competition (asymmetry; size- and space-dependence) are accounted for. Implications of these results are discussed in the context of plant ecology and in more general modelling situations.     

Ungulate browsing in winter reduces the growth of <Emphasis Type="Italic">Fraxinus</Emphasis> and <Emphasis Type="Italic">Acer</Emphasis> saplings in subsequent unbrowsed years

Browsing by ungulates has become a hotly debated issue in many European mountain forests in the past century. Acer pseudoplatanus L. and Fraxinus excelsior L. are broadleaved tree species that are preferentially browsed by roe deer (Capreolus capreolus) in central Europe. We analyzed growth data from shaded saplings of both tree species to quantify the extent to which height growth after game browsing is reduced in subsequent, unbrowsed years in forest stands. Sixty saplings of F. excelsior and A. pseudoplatanus from forest stands at Albisriederberg (Switzerland) were available to us that had been dissected into pieces that then were split in the middle for counting tree rings and assessing ungulate damage. We fitted the von Bertalanffy growth equation to these height growth data and included a reduction factor for winter browsing. Both tree species showed significantly reduced height growth in unbrowsed years after one to several browsing events in winter, and this effect increased with the number of browsing events. Saplings with a high growth rate showed a higher growth reduction. After winter browsing, height growth of A. pseudoplatanus saplings was less affected in unbrowsed years than that of F. excelsior saplings. We conclude that browsed saplings of these species in forest stands are not able to compensate browsing-induced height loss, but that height differences between browsed and unbrowsed saplings probably increase over time. A comparison between our analysis and the parameters estimated using equations published by Eiberle for predicting age at 130 cm height suggests that our parameter values are rather conservative estimates of the growth reduction effect after winter browsing. Neither F. excelsior nor A. pseudoplatanus show a distinct pattern in browsing-induced growth reduction with respect to soil moisture, nutrient level, and altitude. We thus conclude that our results are likely to be valid for a wide range of forested sites.     

Factors affecting estimates of size at age and growth in grey triggerfish Balistes capriscus from the northern Gulf of Mexico

Growth zones in dorsal spines of grey triggerfish Balistes capriscus from the northern Gulf of Mexico were utilized to estimate growth and examine factors that may affect estimates of size at age. Age was estimated from dorsal‐spine sections of 4687 individuals sampled from U.S. waters during 2003–2013, including both fishery‐independent (n = 1312) and fishery‐dependent (n = 3375) samples. Ninety‐six per cent (n = 4498) of these sections were deemed suitable for ageing; average per cent error between two independent readers was 10·8%. Fork length (L_F) ranged from 65 to 697 mm and age estimates from 0 to 14 years. Both sex and sample source (fishery‐independent v. recreational) significantly affected estimated size at age for 2–6 year‐old fish. Data were pooled between sources to fit sex‐specific von Bertalanffy growth functions. Results for the female model were L_∞ = 387 mm L_F, k = 0·52 year^?1 and t₀ = 0·01 year, while for males L_∞ = 405 mm L_F, k = 0·55 year^?1 and t₀ = 0·02 year. These results were significantly different between sexes and indicate clear sexual dimorphism. Thus, growth should be modelled separately by sex when examining population parameters or conducting stock assessment modelling. The positive bias in estimates of size at age computed for recreational v. fishery‐independent samples also has clear implications for stock assessment as growth functions computed with fishery‐dependent samples would tend to overestimate stock productivity.     

Field studies on the ureogenic gulf toadfish in a subtropical bay. I. Patterns of abundance, size composition and growth

Opsanus beta abundance in Biscayne Bay, Florida, varied significantly depending on site and season. O. beta catch per unit effort (cpue) was highest in southern study sites which tended to provide the combination of shallow mean depth, nocturnally high levels of dissolved oxygen, and Thalassia -dominated substrate. Significant seasonal differences in abundance were restricted to the three sites where O. beta cpue tended to be highest; two of these sites shared the same pattern of in crease during the cool-dry season (November-April), followed by a return to lower levels during warm-wet season (May-September). Dramatic seasonal abundance increases could not be ascribed to large influxes of immature fish. Rather, size composition data suggested sexually mature individuals moved seasonally among habitats, possibly for reproductive purposes. Modal analysis of length-frequency data followed by nonlinear regressions yielded preliminary values for the von Bertalanffy growth parameters K and t₀ , given that L_σ = 300 mm. Possible reasons for O. beta growing slower and attaining a smaller maximal size than its temperate congener, O. tau , are discussed.     

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.     

Age,growth and reproduction of Marcusenius pongolensis,Oreochromis mossambicus and Schilbe intermedius in an oligotrophic impoundment in Swaziland

The age, growth and reproductive biology of Marcusenius pongolensis, Oreochromis mossambicus and Schilbe intermedius were investigated in the Mnjoli Dam, Swaziland. Otolith annulus formation occurred in winter for M. pongolensis, and in spring/summer for O. mossambicus and S. intermedius. Maximum ages of 8, 6 and 8 years were recorded for M. pongolensis, O. mossambicus and S. intermedius, respectively. Growth was described by the von Bertalanffy growth model as L_t = 238.73(1 − exp^{−0.27(t+2.27}) mm fork length (FL) for M. pongolensis, L_t = 226.83(1 − exp^{−0.45(t+2.02)}) mm total length (TL) for O. mossambicus, and L_t = 214.59(1 − exp^{−0.60(t+1.20)}) mm FL for S. intermedius. Sexual maturity was estimated for male and female M. pongolensis at 134 mm FL and 119 mm FL, respectively. Marcusenius pongolensis matured within their first year. Female O. mossambicus and S. intermedius matured at 239 mm TL and 205 mm FL, corresponding to 2 and 4 years of age, respectively. Extended spawning periods, with two spawning peaks was observed for M. pongolensis, one in spring (September) and the second in autumn (March) and one peak over late‐summer for S. intermedius.     

Growth and age determination of African savanna elephants

Understanding the population dynamics of savanna elephants depends on estimating population parameters such as the age at first reproduction, calving interval and age-specific survival rates. The generation of these parameters, however, relies on the ability to accurately determine the age of individuals, but a reliable age estimation technique for free-ranging elephants is presently not available. Shoulder heights of elephants were measured in 10 populations in five countries across southern and eastern Africa. Data included shoulder height measurements from two populations where the age of each individual was known (i.e. Addo Elephant National Park, South Africa and Amboseli National Park, Kenya). From the known-age data, Von Bertalanffy growth functions were constructed for both male and female elephants. Savanna elephants were found to attain similar asymptotic shoulder heights in the 10 populations, while individuals in the two known-age populations grew at the same rate. The Von Bertalanffy growth curves allowed for the accurate age estimation of females up to 15 years of age and males up to 36 years of age. The results indicate that shoulder height can serve as an indicator of chronological age for elephants below 15 years of age for females and 36 years of age for males. Ages derived from these growth curves can then be used to generate age-specific population variables, which will help assess the demographic status of savanna elephant populations across Africa.     

Resource polymorphism and diversity of Arctic charr Salvelinus alpinus in a series of isolated lakes

Morphological, dietary and life‐history variation in Arctic charr Salvelinus alpinus were characterized from three geographically proximate, but isolated lakes and one large lake into which they drain in south‐western Alaska. Polymorphism was predicted to occur in the first three lakes because S. alpinus tend to become polymorphic in deep, isolated lakes with few co‐occurring species. Only one morph was evident in the large lake and two of the three isolated lakes. In the third isolated lake, Lower Tazimina Lake, small and large morphs were found, the latter including two forms differing in growth rate. The small morph additionally differed from the two large forms by having more gill rakers and a deeper body than same‐sized individuals of the large morph, consuming more limnetic and fewer benthic resources, having a greater gonado‐somatic index and maturing at a smaller size. The two large forms consumed only slightly different foods (more terrestrial insects were consumed by the medium‐growth form; more snails by the high‐growth form). Trends in consumption of resources with body shape also differed between lakes. Variability in life history of S. alpinus in these Alaskan lakes was as broad as that found elsewhere. This variability is important for understanding lake ecosystems of remote regions where this species is commonly dominant.     

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

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

Growth with seasonally varying temperatures: an expansion of the von Bertalanffy growth model

The von Bertalanffy growth function has limitations for describing the growth of fishes in seasonal climates. In the present work, a new equation is proposed where the growth parameter k is substituted by a function related to monthly water temperatures. The computer program GROWTHS was developed to fit and simulate the growth for seasonally varying temperatures. Examples for natural populations of Barbatula barbatula and Cottus gobio are presented.     

Age and growth of the gulf toadfish Opsanus beta based on otolith increment analysis

In the present study, sagittal otoliths of confirmed male and female specimens of the gulf toadfish Opsanus beta that were collected monthly over the course of a year from Biscayne Bay, Florida, U.S.A. were analysed. The timing and frequency of O. beta spawning seasons are reported by examination of the gonado‐somatic index. The estimated ages of males and females ranged from < 1 year to 6 and 5 years, respectively. Strong sexual dimorphism in growth was apparent with von Bertalanffy parameter estimates for males of L_∞ = 393·8 mm, K = 0·30, t₀ = 0·36 and females of L_∞ = 201·1 mm, K = 0·79, t₀ = 0·47. Comparison with previously published growth trajectories of the more northerly distributed conspecific Opsanus tau showed that O. beta males had a higher growth rate. Female O. beta and O. tau growth trajectories appear similar, with an indication that the former becomes asymptotic at least a year before the latter. Results are discussed in the context of temperature regimes, reproductive energy allocation and waste urea excretion in the two species.