Empirical relationships to estimate asymptotic length, length at first maturity and length at maximum yield per recruit in fishes, with a simple method to evaluate length frequency data

Empirical relationships are presented to estimate in fishes, asymptotic length (L∞) from maximum observed length (L_max), length at first maturity (L_m) from L_∞, life span (t_max) from age at first maturity (t_m), and length at maximum possible yield per recruit (L_opt) from L_∞ and from L_m, respectively. The age at L_opt is found to be a good indicator of generation time in fishes. A spreadsheet containing the various equations can be downloaded from the Internet at http://www.fishbase.org/download as popdynJFB.zip. A simple method is presented for evaluation of length–frequency data in their relationship to L_∞, L_m and L_opt. This can be used to evaluate the quality of the length–frequency sample and the status of the population. Three examples demonstrate the usefulness of this method. 2000 The Fisheries Society of the British Isles     

Length–weight,length–length relationships and length at first maturity of fish species from the Paraná and Uruguay rivers,Argentina

Length–weight relationships (LWR) were estimated for 12 species, representing three orders and eight families of fishes from the Paraná and Uruguay rivers in Argentina: Acestrorhynchus pantaneiro, Cynopotamus argenteus, Pachyurus bonariensis, Pterodoras granulosus, Hypostomus commersoni, Pseudoplatystoma corruscans, Pimelodus maculatus, Parapimelodus valenciennis, Salminus brasiliensis, Prochilodus lineatus, Hoplias malabaricus and Leporinus obtusidens. For the last four species, the length at first spawning (L_m) was estimated using three different methods. The captures were made in shallow river areas during 2005–2013. Significant length–weight relationships were found for all species. None or few LWRs were previously available for these species.     

A new method for estimating age-at-death from the first rib

A new method for estimating adult age-at-death from the first rib was developed as a modification of the Kunos et al. (Am J Phys Anthropol 110 (1999) 303-323) method. Data were collected on three aspects of the first rib (costal face, rib head, and tubercle facet) for 470 known-age males of Balkan ancestry collected as evidence during investigations conducted by the International Criminal Tribunal for the former Yugoslavia (ICTY). Ages-at-death range from 12 to 90 years (mean of 47.7 years). Several variables were extracted from the original study utilizing all three skeletal aspects of the first rib. This list was modified to 11 variables as preliminary tests on seriations of the samples were undertaken. A cumulative probit model with age measured on a log scale was used to calculate the mean and standard deviation of the ages-of-transition for each component. Multivariate analysis of the three components was also performed. The lowest correlation (r = 0.079, controlling for age) was between the geometric shape of the costal face and the surface texture of the tubercle facet. Assuming a correlation of zero, these two traits were used to calculate the highest posterior density regions for estimating individual ages-at-death. Age-at-death estimates generated from 50 and 95% posterior density regions indicate that this method captures age-related change reaching the ninth decade. The Bayesian statistical approach used here produced a valuable and promising new method for estimating age-at-death. Additional research is necessary to determine if these highest posterior density regions produce results highly correlated with age in other samples and its applicability to females.     

Techniques for estimating allometric equations.

Morphologists have long been aware that differential size relationships of variables can be fo great value when studying shape. Allometric patterns have been the basis of many interpretations of adaptations, biomechanisms, and taxonomies. It is of importance that the parameters of the allometric equation be as accurate estimates as possible since they are so commonly used in such interpretations. Since the error term may come into the allometric relation either exponentially or additively, there are at least two methods of estimating the parameters of the allometric equation. That most commonly used assumes exponentiality of the error term, and operates by forming a linear function by a logarithmic transformation and then solving by the method of ordinary least squares. On the other hand, if the rrror term comes into the equation in an additive way, a nonlinear method may be used, searching the parameter space for those parameters which minimize the sum of squared residuals. Study of data on body weight and metabolism in birds explores the issues involved in discriminating between the two models by working through a specific example and shows that these two methods of estimation can yield highly different results. Not only minimizing the sum of squared residuals, but also the distribution and randomness of the residuals must be considered in determing which model more precisely estimates the parameters. In general there is no a priori way to tell which model will be best. Given the importance often attached to the parameter estimates, it may be well worth considerable effort to find which method of solution is appropriate for a given set of data.     

Length–weight relationships and length at first maturity for fish species in the upper Miranda River,southern Pantanal wetland,Brazil

Length–weight relationships (LWR) were estimated for 17 species and total length at first maturity (L50) for three species of freshwater fishes from the Miranda River, southern Pantanal, Brazil. The b values were compared for some species in the Paraguay River basin with the northern (Cuiabá River) part of the basin; differences in length–weight relationships were significantly different for Pseudoplatystoma corruscans, P. reticulatum (syn. P. fasciatum). First references on L50 and LWR are presented for two and eight fish species, respectively, as well as the new maximum total length for two species.     

The interaction of season length and development time alters size at maturity

An end-of-season penalty, with late-maturing individuals being smaller than early-maturing individuals, has been observed in a variety of univoltine terrestrial arthropods. The current study extends these observations, utilizing multiple populations of a single sexually dimorphic species to examine the ecological correlates and fitness consequences of late maturation at a small size. The orb-weaving spider, Nephila clavipes, inhabits a broad range of habitats that vary from mild to strong seasonality. Because males mature several instars earlier than females, they can reach maturity much earlier in the growing season. Within a cohort, I found that female size at maturity was negatively correlated with timing of maturation in strongly seasonal sites. At a less seasonal site, there was no correlation between female size and timing of maturation within a cohort. In most populations studied, male size was not correlated with the timing of maturation within a cohort. Within populations in strongly seasonal sites, late-maturing females had reduced fecundity. The probability of copulation, survivorship from maturity to first clutch, clutch size relative to female size, and the number of possible clutches were all reduced with delayed maturation. The probability of pre-reproductive death for late-maturing females was strongly affected by stochasticity in the timing of the end of the growing season. Received: 30 December 1998 / Accepted: 1 September 1999     

Testing for bias in weighted estimating equations

It is very common in regression analysis to encounter incompletely observed covariate information. A recent approach to analyse such data is weighted estimating equations (Robins, J. M., Rotnitzky, A. and Zhao, L. P. (1994), JASA, 89, 846-866, and Zhao, L. P., Lipsitz, S. R. and Lew, D. (1996), Biometrics, 52, 1165-1182). With weighted estimating equations, the contribution to the estimating equation from a complete observation is weighted by the inverse of the probability of being observed. We propose a test statistic to assess if the weighted estimating equations produce biased estimates. Our test statistic is similar to the test statistic proposed by DuMouchel and Duncan (1983) for weighted least squares estimates for sample survey data. The method is illustrated using data from a randomized clinical trial on chemotherapy for multiple myeloma.     

Traditional approaches to estimate length at first maturity (L50) retrieve better results than alternative ones in a Neotropical heptapterid

Length at first maturity (L₅₀) is an important tool for the management and conservation of fish populations. Traditional approaches based on macroscopic and microscopic maturity staging exhibit high accuracy and precision, while alternative approaches (e.g., I_g-based staging, stanza changing point) are less resource-demanding. Herein, we compare four approaches to estimate L₅₀ in a population of the heptapterid Rhamdioglanis transfasciatus from Atlantic Forest streams. Fish were sampled monthly during a year by using electrofishing. We measured the length (cm), mass (g), and gonad mass (g) of each specimen, then classified their maturity status macroscopically and microscopically. Alternative approaches were strongly discordant from traditional ones. Logistic curves considering mature individuals as those displaying at least 1% of the maximum I_g in the sample greatly underestimated L₅₀ for females and overestimated L₅₀ for males. The stanza changing point derived from the polyphasic growth model underestimated L₅₀ in both cases. Despite the increasing development of less onerous approaches, it seems that they are not suitable for all fish populations and the requirements to use such approaches demand further investigation.     

Length–weight relationships and length at first maturity for nine fish species of floodplain lakes in Central Amazon (Amazon Basin,Brazil)

Length–weight relationships are presented for nine fish species from the floodplain lakes in the Central Amazon (Amazon Basin, Brazil). The parameter slope b values in the length–weight relationships ranged from 2.33 to 3.28 for grouped sexes, and from 2.7 to 3.61 for separated sexes. Differences between sexes were verified in three species. Sizes at first sexual maturation ranged from 9.14 to 23.97. This study provides a new reference for the length–weight relationships of six species.     

Length–weight and length–length relationships and length at first maturity for freshwater fish species of the Salto Santiago Reservoir,Iguaçu River Basin,Brazil

Length‐weight (LWR) and length‐length (LLR) relationships were estimated for 20 species and lengths at first maturity (L₅₀) for six species of freshwater fishes caught in the Salto Santiago Reservoir, Iguaçu River Basin, Brazil. In nine species significant differences were found in the LWR between sexes. Average b‐value for species with no differences between sexes in LWR was 3.12 (SE = ±0.05). Average b‐value in LLR was 0.823. First references on LWRs and L₅₀ are presented for 13 and four fish species, respectively, as well as the new maximum total lengths for eight species.     

Regression equations for estimating stature from long bones of early Holocene European samples

Regression equations for estimating living stature from long bone lengths have been calibrated on a sample of European Neolithic skeletons (33 males and 27 females) by using both least-squares (model I) and major-axis (model II) regression techniques. Stature estimates of the skeletal sample have been made by means of Fully's anatomical method, a procedure based on the sum of all osseous components of height, providing the best approximations to the actual stature. The calculated equations have been tested, along with those generally used to predict stature of earlier European remains, on a small, well-preserved sample including Late Upper Paleolithic, Mesolithic, and Neolithic skeletons. The results indicate that the model II equations are particularly useful when very short or very tall individuals are involved and, at the same time, are among the best predictors of stature in less extreme conditions. © 1996 Wiley-Liss, Inc.     

A note on deletion diagnostics for estimating equations

We describe an algorithm based upon the Sherman–Morrison–Woodburyformula for the inversion of matrices with special structurethat occur in formulae for deletion diagnostics. Substantialcomputational savings relative to a method based upon Cholesky'sdecomposition are illustrated. The result has broad applicationto regression diagnostics for clustered data.     

The Gill-Oxygen Limitation Theory and size at maturity/maximum size relationships for salmonid populations occupying flowing waters

The slowing of growth as fish age has long been believed to be related to energy expenditure for maturation, and this rationalization has been used to explain why, across nearly all fish species, the relationship between size at first maturity (L_m) and maximum (L_max) or asymptotic length (L_∞) is relatively constant. In contrast, the Gill-Oxygen Limitation Theory (GOLT) postulates that (a) fish growth slows because as they grow, their two-dimensional ability to extract oxygen from the water diminishes relative to their three-dimensional weight gain, and (b) they can only invest energy for maturation if oxygen supply at their size at first maturity (Q_m) exceeds that needed for maintenance metabolism (Q_∞). It has been reported previously across dozens of marine fish species that the relationship between Q_m and Q_∞ is linear and, further, it can be mathematically converted to L_m vs. L_∞ by raising both terms to the power of D (the gill surface factor), resulting in a slope of 1.36. If the GOLT is universal, a similar slope should exist for L_m^D vs. L_∞^D relationships for freshwater species across multiple individual populations that reside in disparate habitats, although to our knowledge this has never been evaluated. For analysis, we used existing data from previous studies conducted on 51 stream-dwelling populations of redband trout Oncorhynchus mykiss gairdneri, Yellowstone cutthroat trout O. clarkii bouvieri and mountain whitefish Prosopium williamsoni. The resulting L_m^D vs. L_∞^D slopes combining all data points (1.35) or for all species considered separately (range = 1.29–1.40) were indeed equivalent to the slope originally produced for the marine species from which the GOLT-derived relationship was first reported. We briefly discuss select papers both supporting and resisting various aspects of the GOLT, note that it could potentially explain shrinking sizes of marine fish, and call for more concerted research efforts combining laboratory and field expertise in fish growth research.     

Regional differences in length at sexual maturity for female blue whales based on recovered Soviet whaling data

New blue whale ovarian corpora data from illegal Soviet catches in the Southern Hemisphere and northern Indian Ocean were recovered from the original logbooks. Catches north of 52°S were assumed to be pygmy blue whales ( Balaenoptera musculus brevicauda , n = 1,272); those south of 56°S were assumed to be Antarctic (true) blue whales ( B. m. intermedia , n = 153). Three probable Antarctic blue whales north of 52°S were excluded. Lengths at which 50% and 95% of females become sexually mature ( L ₅₀ and L ₉₅) were estimated from a Bayesian logistic model. These estimates are more precise than previous Japanese estimates because Soviet catches below the legal minimum of 70 ft (21.3 m) were 32 times greater. For pygmy blue whales L ₅₀ was 19.2 m (95% interval 19.1–19.3 m) and L ₉₅ was 20.5 m (95% interval 20.4–20.7 m). Antarctic L ₅₀ (23.4 m, 95% interval 22.9–23.9 m) was much longer than L ₅₀ for pygmy blue whale regions (18.4–19.9 m). The median L ₅₀ for the northern Indian Ocean was 0.5–0.6 m shorter than for pygmy blue whales from other regions; although statistically significant, these small length differences provide little support for northern Indian Ocean blue whales being a separate subspecies, B. m. indica .     

Variation in length and age at sexual maturity of Atlantic groundfish: a reply

Synopsis The assertion has been made by Halliday (1987) that trends in size and age at maturity of Atlantic groundfish published by Beacham (1983a, b, c, d, e, f) are artifacts induced by errors in determining the sex of an individual, distinguishing between immature and mature fish, sampling fish outside of the regular spawning season, and by nonrandom sampling of the population. In particular, Halliday asserts that for the Atlantic argentine,Argentina silus analysis, the conclusions of Beacham (1983a) that: (1) median length at sexual maturity declined over time; and (2) males matured at older ages than did females are invalid owing to biases in both sampling and analysis. In fact, if some of the biases indicated by Halliday were significant, then the decline in median length at sexual maturity is enhanced and the conclusions of Beacham (1983a) reinforced. Size and age at sexual maturity are dynamic characters in many vertebrate populations, and the fact that they should change for Atlantic groundfish should not be surprising given variable exploitation patterns in the fisheries since 1960.