A note on the von Bertalanffy growth function concerning the allocation of surplus energy to reproduction

We propose an extended form of the von Bertalanffy growth function (VBGF), where the allocation of surplus energy to reproduction is considered. Any function can be used in our model to describe the ratio of energy allocation for reproduction to that for somatic growth. As an example, two models for energy allocation were derived: a step-function and a logistic function. The extended model can jointly describe growth in adult and juvenile stages. The change in growth rate between the two stages can be either gradual or steep; the latter gives a biphasic VBGF. The results of curve fitting indicated that a consideration of reproductive energy is meaningful for model extension. By controlling parameter values, our comprehensive model gives various growth curve shapes ranging from indeterminate to determinate growth. An increase in the number of parameters is unavoidable in practical applications of this new model. Additional information on reproduction will improve the reliability of model estimates.

Age and growth of spiny dogfish (Squalus acanthias) in the Gulf of Alaska: analysis of alternative growth models

Ten growth models were fitted to age and growth data for spiny dogfish (Squalus acanthias) in the Gulf of Alaska. Previous studies of spiny dogfish growth have all fitted the t0 formulation of the von Bertalanffy model without examination of alternative models. Among the alternatives, we present a new two-phase von Bertalanffy growth model formulation with a logistically scaled k parameter and which estimates L0. A total of 1602 dogfish were aged from opportunistic collections with longline, rod and reel, set net, and trawling gear in the eastern and central Gulf of Alaska between 2004 and 2007. Ages were estimated from the median band count of three independent readings of the second dorsal spine plus the estimated number of worn bands for worn spines. Owing to a lack of small dogfish in the samples, lengths at age of small individuals were back-calculated from a subsample of 153 dogfish with unworn spines. The von Bertalanffy, two-parameter von Bertalanffy, two-phase von Bertalanffy, Gompertz, two-parameter Gompertz, and logistic models were fitted to length-at-age data for each sex separately, both with and without back-calculated lengths at age. The two-phase von Bertalanffy growth model produced the statistically best fit for both sexes of Gulf of Alaska spiny dogfish, resulting in L∞ = 87.2 and 102.5 cm and k= 0.106 and 0.058 for males and females, respectively.

An allometric smoothing function to describe the relation between otolith and somatic growth over the lifespan of walleye pollock (Theragra chalcogramma)

We propose a new equation to describe the relation between otolith length (OL) and somatic length (fork length [FL]) of fish for the entire lifespan of the fish. The equation was developed by applying a mathematical smoothing method based on an allometric equation with a constant term for walleye pollock (Theragra chalcogramma) —a species that shows an extended longevity (>20 years). The most appropriate equation for defining the relation between OL and FL was a four-phase allometric smoothing function with three inflection points. The inflection points correspond to the timing of settlement of walleye pollock, changes in sexual maturity, and direction of otolith growth. Allometric smoothing functions describing the relation between short otolith radius and FL, long otolith radius and FL, and FL and body weight were also developed. The proposed allometric smoothing functions cover the entire lifespan of walleye pollock. We term these equations “allometric smoothing functions for otolith and somatic growth over the lifespan of walleye pollock.”

Growth, mortality, and hatchdate distributions of larval and juvenile spotted seatrout (Cynoscion nebulosus) in Florida Bay, Everglades National Park

Life history aspects of larval and, mainly, juvenile spotted seatrout (Cynoscion nebulosus) were studied in Florida Bay, Everglades National Park, Florida. Collections were made in 1994−97, although the majority of juveniles were collected in 1995. The main objective was to obtain life history data to eventually develop a spatially explicit model and provide baseline data to understand how Everglades restoration plans (i.e. increased freshwater flows) could influence spotted seatrout vital rates. Growth of larvae and juveniles (<80 mm SL) was best described by the equation loge standard length = –1.31 + 1.2162 (loge age). Growth in length of juveniles (12–80 mm SL) was best described by the equation standard length = –7.50 + 0.8417 (age). Growth in wet weight of juveniles (15–69 mm SL) was best described by the equation loge wet-weight = –4.44 + 0.0748 (age). There were no significant differences in juvenile growth in length of spotted seatrout in 1995 between three geographical subdivisions of Florida Bay: central, western, and waters adjacent to the Gulf of Mexico. We found a significant difference in wet-weight for one of six cohorts categorized by month of hatchdate in 1995, and a significant difference in length for another cohort. Juveniles (i.e. survivors) used to calculate weekly hatchdate distributions during 1995 had estimated spawning times that were cyclical and protracted, and there was no correlation between spawning and moon phase. Temperature influenced otolith increment widths during certain growth periods in 1995. There was no evidence of a relationship between otolith growth rate and temperature for the first 21 increments. For increments 22–60, otolith growth rates decreased with increasing age and the extent of the decrease depended strongly in a quadratic fashion on the temperature to which the fish was exposed. For temperatures at the lower and higher range, increment growth rates were highest. We suggest that this quadratic relationship might be influenced by an environmental factor other than temperature. There was insufficient information to obtain reliable inferences on the relationship of increment growth rate to salinity.

Modeling red sea urchin (Strongylocentrotus franciscanus) growth using six growth functions

The growth of red sea urchins (Strongylocentrotus franciscanus) was modeled by using tag-recapture data from northern California. Red sea urchins (n=211) ranging in test diameter from 7 to 131 mm were examined for changes in size over one year. We used the function Jt+1 = Jt + f(Jt) to model growth, in which Jt is the jaw size (mm) at tagging, and Jt+1 is the jaw size one year later. The function f(Jt), represents one of six deterministic models: logistic dose response, Gaussian, Tanaka, Ricker, Richards, and von Bertalanffy with 3, 3, 3, 2, 3, and 2 minimization parameters, respectively. We found that three measures of goodness of fi t ranked the models similarly, in the order given. The results from these six models indicate that red sea urchins are slow growing animals (mean of 7.2 ±1.3 years to enter the fishery). We show that poor model selection or data from a limited range of urchin sizes (or both) produces erroneous growth parameter estimates and years-to-fishery estimates. Individual variation in growth dominated spatial variation at shallow and deep sites (F=0.246, n=199, P=0.62). We summarize the six models using a composite growth curve of jaw size, J, as a function of time, t: J = A(B – e–Ct) + Dt, in which each model is distinguished by the constants A, B, C, and D. We suggest that this composite model has the flexibility of the other six models and could be broadly applied. Given the robustness of our results regarding the number of years to enter the fishery, this information could be incorporated into future fishery management plans for red sea urchins in northern California.

Preliminary estimation of growth in Cheimerius nufar (Sparidae) from southern Mozambique

On the basis of length-frequency data collected in 1987, by the hook-and-line fishery research program, preliminary estimates of the parameters of a seasonally oscillating version of the von Bertalanffy equation for Cheimerius nufar (Ehrenberg, 1830) were obtained, i.e., TL∞=70 cm and K=0.17 yearˉ¹. A discussion of these results is provided.