Maximum likelihood estimation of mortality and growth with individual variability from multiple length-frequency data

We consider estimation of mortality rates and growth parameters from length-frequency data of a fish stock and derive the underlying length distribution of the population and the catch when there is individual variability in the von Bertalanffy growth parameter L-infinity. The model is flexible enough to accommodate 1) any recruitment pattern as a function of both time and length, 2) length-specific selectivity, and 3) varying fishing effort over time. The maximum likelihood method gives consistent estimates, provided the underlying distribution for individual variation in growth is correctly specified. Simulation results indicate that our method is reasonably robust to violations in the assumptions. The method is applied to tiger prawn data (Penaeus semisulcatus) to obtain estimates of natural and fishing mortality.

Estimation of growth parameters from multiple-recapture data

This article develops a method for analysis of growth data with multiple recaptures when the initial ages for all individuals are unknown. The existing approaches either impute the initial ages or model them as random effects. Assumptions about the initial age are not verifiable because all the initial ages are unknown. We present an alternative approach that treats all the lengths including the length at first capture as correlated repeated measures for each individual. Optimal estimating equations are developed using the generalized estimating equations approach that only requires the first two moment assumptions. Explicit expressions for estimation of both mean growth parameters and variance components are given to minimize the computational complexity. Simulation studies indicate that the proposed method works well. Two real data sets are analyzed for illustration, one from whelks (Dicathais aegaota) and the other from southern rock lobster (Jasus edwardsii) in South Australia.

Growth curves with explanatory variables and estimation of the effect of tagging

The von Bertalanffy growth model is extended to incorporate explanatory variables. The generalized model includes the switched growth model and the seasonal growth model as special cases, and can also be used to assess the tagging effect on growth. Distribution-free and consistent estimating functions are constructed for estimation of growth parameters from tag-recapture data in which age at release is unknown. This generalizes the work of James (1991, Biometrics 47 1519-1530) who considered the classical model and allowed for individual variability in growth. A real dataset from barramundi (Lates calcarifer) is analysed to estimate the growth parameters and possible effect of tagging on growth.

From a tree to a stand in Finnish boreal forests: biomass estimation and comparison of methods

There is an increasing need to compare the results obtained with different methods of estimation of tree biomass in order to reduce the uncertainty in the assessment of forest biomass carbon. In this study, tree biomass was investigated in a 30-year-old Scots pine (Pinus sylvestris) (Young-Stand) and a 130-year-old mixed Norway spruce (Picea abies)-Scots pine stand (Mature-Stand) located in southern Finland (61º50' N, 24º22' E). In particular, a comparison of the results of different estimation methods was conducted to assess the reliability and suitability of their applications. For the trees in Mature-Stand, annual stem biomass increment fluctuated following a sigmoid equation, and the fitting curves reached a maximum level (from about 1 kg/yr for understorey spruce to 7 kg/yr for dominant pine) when the trees were 100 years old. Tree biomass was estimated to be about 70 Mg/ha in Young-Stand and about 220 Mg/ha in Mature-Stand. In the region (58.00-62.13 ºN, 14-34 ºE, ≤ 300 m a.s.l.) surrounding the study stands, the tree biomass accumulation in Norway spruce and Scots pine stands followed a sigmoid equation with stand age, with a maximum of 230 Mg/ha at the age of 140 years. In Mature-Stand, lichen biomass on the trees was 1.63 Mg/ha with more than half of the biomass occurring on dead branches, and the standing crop of litter lichen on the ground was about 0.09 Mg/ha. There were substantial differences among the results estimated by different methods in the stands. These results imply that a possible estimation error should be taken into account when calculating tree biomass in a stand with an indirect approach.

Improved estimation of size-transition matrices using tag-recapture data

We derive a new method for determining size-transition matrices (STMs) that eliminates probabilities of negative growth and accounts for individual variability. STMs are an important part of size-structured models, which are used in the stock assessment of aquatic species. The elements of STMs represent the probability of growth from one size class to another, given a time step. The growth increment over this time step can be modelled with a variety of methods, but when a population construct is assumed for the underlying growth model, the resulting STM may contain entries that predict negative growth. To solve this problem, we use a maximum likelihood method that incorporates individual variability in the asymptotic length, relative age at tagging, and measurement error to obtain von Bertalanffy growth model parameter estimates. The statistical moments for the future length given an individual’s previous length measurement and time at liberty are then derived. We moment match the true conditional distributions with skewed-normal distributions and use these to accurately estimate the elements of the STMs. The method is investigated with simulated tag–recapture data and tag–recapture data gathered from the Australian eastern king prawn (Melicertus plebejus).

Population projection for Namibia 2011-2020 - With an estimation of the impact of HIV/AIDS

This study presents a population projection for Namibia for years 2011–2020. In many countries of sub-Saharan Africa, including Namibia, the population growth is still continuing even though the fertility rates have declined. However, many of these countries suffer from a large HIV epidemic that is slowing down the population growth. In Namibia, the epidemic has been severe. Therefore, it is important to assess the effect of HIV/AIDS on the population of Namibia in the future. Demographic research on Namibia has not been very extensive, and data on population is not widely available. According to the studies made, fertility has been shown to be generally declining and mortality has been significantly increasing due to AIDS. Previous population projections predict population growth for Namibia in the near future, yet HIV/AIDS is affecting the future population developments. For the projection constructed in this study, data on population is taken from the two most recent censuses, from 1991 and 2001. Data on HIV is available from HIV Sentinel Surveys 1992–2008, which test pregnant women for HIV in antenatal clinics. Additional data are collected from different sources and recent studies. The projection is made with software (EPP and Spectrum) specially designed for developing countries with scarce data. The projection includes two main scenarios which have different assumptions concerning the development of the HIV epidemic. In addition, two hypothetical scenarios are made: the first considering the case where HIV epidemic would never have existed and the second considering the case where HIV treatment would never have existed. The results indicate population growth for Namibia. Population in the 2001 census was 1.83 million and is projected to result in 2.38/2.39 million in 2020 in the first two scenarios. Without HIV, population would be 2.61 million and without treatment 2.30 million in 2020. Urban population is growing faster than rural. Even though AIDS is increasing mortality, the past high fertility rates still keep young adult age groups quite large. The HIV epidemic shows to be slowing down, but it is still increasing the mortality of the working-aged population. The initiation of HIV treatment in 2004 in the public sector seems to have had an effect on many projected indicators, diminishing the impact of HIV on the population. For example, the rise of mortality is slowing down.

Estimation of year class abundance and mortality of yellowfin tuna in the Eastern Tropical Pacific

ENGLISH: Age composition of catch, and growth rate, of yellowfin tuna have been estimated by Hennemuth (1961a) and Davidoff (1963). The relative abundance and instantaneous total mortality rate of yellowfin tuna during 1954-1959 have been estimated by Hennenmuth (1961b). It is now possible to extend this work, because more data are available; these include data for 1951-1954, which were previously not available, and data for 1960-1962, which were collected subsequent to Hennemuth's (1961b) publication. In that publication, Hennemuth estimated the total instantaneous mortality rate (Z) during the entire time period a year class is present in the fishery following full recruitment. However, this method may lead to biased estimates of abundance, and hence mortality rates, because of both seasonal migrations into or out of specific fishing areas and possible seasonal differences in availability or vulnerability of the fish to the fishing gear. Schaefer, Chatwin and Broadhead (1961) and Joseph etl al. (1964) have indicated that seasonal migrations of yellowfin occur. A method of estimating mortality rates which is not biased by seasonal movements would be of value in computations of population dynamics. The method of analysis outlined and used in the present paper may obviate this bias by comparing the abundance of an individual yellowfin year class, following its period of maximum abundance, in an individual area during a specific quarter of the year with its abundance in the same area one year later. The method was suggested by Gulland (1955) and used by Chapman, Holt and Allen (1963) in assessing Antarctic whale stocks. This method, and the results of its use with data for yellowfin caught in the eastern tropical Pacific from 1951-1962 are described in this paper. SPANISH: La composición de edad de la captura, y la tasa de crecimiento del atún aleta amarilla, han sido estimadas por Hennemuth (1961a) y Davidoff (1963). Hennemuth (1961b), estimó la abundancia relativa y la tasa de mortalidad total instantánea del atún aleta amarilla durante 1954-1959. Se puede ampliar ahora, este trabajo, porque se dispone de más datos; éstos incluyen datos de 1951 1954, de los cuales no se disponía antes, y datos de 1960-1962 que fueron recolectados después de la publicación de Hennemuth (1961b). En esa obra, Hennemuth estimó la tasa de mortalidad total instantánea (Z) durante todo el período de tiempo en el cual una clase anual está presente en la pesquería, consecutiva al reclutamiento total. Sin embargo, este método puede conducir a estimaciones con bias (inclinación viciada) de abundancia, y de aquí las tasas de mortalidad, debidas tanto a migraciones estacionales dentro o fuera de las áreas determinadas de pesca, como a posibles diferencias estacionales en la disponibilidad y vulnerabilidad de los peces al equipo de pesca. Schaefer, Chatwin y Broadhead (1961) y Joseph et al. (1964) han indicado que ocurren migraciones estacionales de atún aleta amarilla. Un método para estimar las tasas de mortalidad el cual no tuviera bias debido a los movimientos estacionales, sería de valor en los cómputos de la dinámica de las poblaciones. El método de análisis delineado y usado en el presente estudio puede evitar este bias al comparar la abundancia de una clase anual individual de atún aleta amarilla, subsecuente a su período de abundancia máxima en un área individual, durante un trimestre específico del año, con su abundancia en la misma área un año más tarde. Este método fue sugerido por Gulland (1955) y empleado por Chapman, Holt y Allen (1963) en la declaración de los stocks de la ballena antártica. Este método y los resultados de su uso, en combinación con los datos del atún aleta amarilla capturado en el Pacífico oriental tropical desde 1951-1962, son descritos en este estudio.

Validation of daily growth increments and estimation of growth rates of larval and early juvenile black skipjack, Euthynnus lineatus, using otoliths

ENGLISH: Increments in otoliths (sagittae) were examined, using light and scanning electron microscopy, to determine ages and estimate growth rates of larval and early-juvenile black skipjack, Euthynnus lineatus. Larvae and juveniles were collected between 1987 and 1989 from coastal waters of Panama in the eastern Pacific Ocean. Results from a laboratory experiment indicated that immersion for 6 and 12 hours in a 200 mg/L solution of tetracycline hydrochloride adequately marks otoliths and that increments are formed daily in the sagittae of postflexion larvae and early juveniles. Further, survival rates of tetracycline-treated fish were not significantly different from those of control fish. Growth rates were derived from length-age relationships of 218 field-collected specimens ranging in size from 5.7 to 20.3 mm SL. A growth rate of 0.70 mm/d was estimated from the weighted regression of standard length on age for all specimens. This rate lies within the range reported for larvae and early juveniles of other species of subtropical and tropical scombrids. Growth rates of postflexion larvae and early juveniles were not significantly different between the rainy season in July-August 1988 and the dry, upwelling season in January-February 1989. Growth was, however, significantly more variable for older individuals in July-August than in January-February, and may correspond, in part, to seasonal patchiness of prey. The growth rates of the otoliths relative to fish length were also not significantly different between seasons; however, the otoliths were larger relative to the lengths of fish collected in the rainy season, which may reflect slower growth during earlier larval stages. SPANISH: Se examinaron incrementos en otolitos (ságitas), usando microscopia de luz y de barrido electrónico, a fin de determinar la edad y estimar las tasas de crecimiento de barriletes negros, Euthynnus lineatus, larvales y juveniles tempranos. Entre 1987 y 1989 se capturaron larvas y juveniles en las aguas costeras de Panamá en el Océano Pacífico oriental. Los resultados de un experimento de laboratorio indicaron que una inmersión de 6 a 12 horas de duración en una solución de 200 mg/L de hidrocloro de tetraciclina marca los otolitos adecuadamente y que los incrementos se forman a diario en las ságitas de larvas en postflexión y juveniles tempranos. Además, las tasas de supervivencia de los peces tratados con tetraciclina no fueron significativamente diferentes a aquellas de los peces de control. Se calcularon las tasas de crecimiento a partir de las relaciones de talla-edad de 218 especímenes de TE entre 5.7 y 20.3 mm capturados en el mar. Se estimó.una tasa de crecimiento de 0.70 mm/día a partir de la regresión ponderada de talla estándar sobre edad para todos los especímenes. Esta tasa cae dentro del rango reportado para larvas y juveniles tempranos de otras especies de escómbridos subtropicales y tropicales. Las tasas de crecimiento de larvas en postflexión y juveniles tempranos no fueron significativamente diferentes entre la temporada de lluvias en julio-agosto de 1988 y la temporada de sequía y afloramiento en enero-febrero de 1989. Sin emoargo, el crecimiento fue significativamente más variable para los individuos de mayor edad en julio-agosto que en enero-febrero, y quizás corresponda parcialmente a la irregularidad temporal de la abundancia de presas. Las tasas de crecimiento de los otolitos en relación a la talla de los peces tampoco fueron significativamente diferentes entre temporadas; sin embargo, los otolitos eran más grandes en relación a la talla en peces capturados en la temporada de lluvias, lo cual podría reflejar crecimiento más lento durante las etapas larvales más tempranas. (PDF contains 42 pages.)

Production of the chironomids of the Uchinsk Reservoir. [Translation of: Methods for the estimation of production of aquatic animals. (Handbook and papers) (ed. G. G. Vinberg) p226-239. Minsk, Vysheishaya Shkola, 1968]

The method of E.V. Borutski was used for determining the production of chironomids, that is, the dynamics of the number and biomass of the larvae were analysed, their death, a calculation of emergence and the number of deposited egg layings was carried out. In addition to the method of Borutski, the authors also calculated the seasonal dynamics of the number of larvae of the younger age stages in the microbenthos.

A-SCALA: an age-structured statistical catch-at-length analysis for assessing tuna stocks in the eastern tropical Pacific Ocean

English: We describe an age-structured statistical catch-at-length analysis (A-SCALA) based on the MULTIFAN-CL model of Fournier et al. (1998). The analysis is applied independently to both the yellowfin and the bigeye tuna populations of the eastern Pacific Ocean (EPO). We model the populations from 1975 to 1999, based on quarterly time steps. Only a single stock for each species is assumed for each analysis, but multiple fisheries that are spatially separate are modeled to allow for spatial differences in catchability and selectivity. The analysis allows for error in the effort-fishing mortality relationship, temporal trends in catchability, temporal variation in recruitment, relationships between the environment and recruitment and between the environment and catchability, and differences in selectivity and catchability among fisheries. The model is fit to total catch data and proportional catch-at-length data conditioned on effort. The A-SCALA method is a statistical approach, and therefore recognizes that the data collected from the fishery do not perfectly represent the population. Also, there is uncertainty in our knowledge about the dynamics of the system and uncertainty about how the observed data relate to the real population. The use of likelihood functions allow us to model the uncertainty in the data collected from the population, and the inclusion of estimable process error allows us to model the uncertainties in the dynamics of the system. The statistical approach allows for the calculation of confidence intervals and the testing of hypotheses. We use a Bayesian version of the maximum likelihood framework that includes distributional constraints on temporal variation in recruitment, the effort-fishing mortality relationship, and catchability. Curvature penalties for selectivity parameters and penalties on extreme fishing mortality rates are also included in the objective function. The mode of the joint posterior distribution is used as an estimate of the model parameters. Confidence intervals are calculated using the normal approximation method. It should be noted that the estimation method includes constraints and priors and therefore the confidence intervals are different from traditionally calculated confidence intervals. Management reference points are calculated, and forward projections are carried out to provide advice for making management decisions for the yellowfin and bigeye populations. Spanish: Describimos un análisis estadístico de captura a talla estructurado por edad, A-SCALA (del inglés age-structured statistical catch-at-length analysis), basado en el modelo MULTIFAN- CL de Fournier et al. (1998). Se aplica el análisis independientemente a las poblaciones de atunes aleta amarilla y patudo del Océano Pacífico oriental (OPO). Modelamos las poblaciones de 1975 a 1999, en pasos trimestrales. Se supone solamente una sola población para cada especie para cada análisis, pero se modelan pesquerías múltiples espacialmente separadas para tomar en cuenta diferencias espaciales en la capturabilidad y selectividad. El análisis toma en cuenta error en la relación esfuerzo-mortalidad por pesca, tendencias temporales en la capturabilidad, variación temporal en el reclutamiento, relaciones entre el medio ambiente y el reclutamiento y entre el medio ambiente y la capturabilidad, y diferencias en selectividad y capturabilidad entre pesquerías. Se ajusta el modelo a datos de captura total y a datos de captura a talla proporcional condicionados sobre esfuerzo. El método A-SCALA es un enfoque estadístico, y reconoce por lo tanto que los datos obtenidos de la pesca no representan la población perfectamente. Además, hay incertidumbre en nuestros conocimientos de la dinámica del sistema e incertidumbre sobre la relación entre los datos observados y la población real. El uso de funciones de verosimilitud nos permite modelar la incertidumbre en los datos obtenidos de la población, y la inclusión de un error de proceso estimable nos permite modelar las incertidumbres en la dinámica del sistema. El enfoque estadístico permite calcular intervalos de confianza y comprobar hipótesis. Usamos una versión bayesiana del marco de verosimilitud máxima que incluye constreñimientos distribucionales sobre la variación temporal en el reclutamiento, la relación esfuerzo-mortalidad por pesca, y la capturabilidad. Se incluyen también en la función objetivo penalidades por curvatura para los parámetros de selectividad y penalidades por tasas extremas de mortalidad por pesca. Se usa la moda de la distribución posterior conjunta como estimación de los parámetros del modelo. Se calculan los intervalos de confianza usando el método de aproximación normal. Cabe destacar que el método de estimación incluye constreñimientos y distribuciones previas y por lo tanto los intervalos de confianza son diferentes de los intervalos de confianza calculados de forma tradicional. Se calculan puntos de referencia para el ordenamiento, y se realizan proyecciones a futuro para asesorar la toma de decisiones para el ordenamiento de las poblaciones de aleta amarilla y patudo.

Estimation of the growth curve parameters in Macrobrachium rosenbergii

Growth is one of the most important characteristics of cultured species. The objective of this study was to determine the fitness of linear, log linear, polynomial, exponential and Logistic functions to the growth curves of Macrobrachium rosenbergii obtained by using weekly records of live weight, total length, head length, claw length, and last segment length from 20 to 192 days of age. The models were evaluated according to the coefficient of determination (R2), and error sum off square (ESS) and helps in formulating breeders in selective breeding programs. Twenty full-sib families consisting 400 PLs each were stocked in 20 different hapas and reared till 8 weeks after which a total of 1200 animals were transferred to earthen ponds and reared up to 192 days. The R2 values of the models ranged from 56 – 96 in case of overall body weight with logistic model being the highest. The R2 value for total length ranged from 62 to 90 with logistic model being the highest. In case of head length, the R2 value ranged between 55 and 95 with logistic model being the highest. The R2 value for claw length ranged from 44 to 94 with logistic model being the highest. For last segment length, R2 value ranged from 55 – 80 with polynomial model being the highest. However, the log linear model registered low ESS value followed by linear model for overall body weight while exponential model showed low ESS value followed by log linear model in case of head length. For total length the low ESS value was given by log linear model followed by logistic model and for claw length exponential model showed low ESS value followed by log linear model. In case of last segment length, linear model showed lowest ESS value followed by log linear model. Since, the model that shows highest R2 value with low ESS value is generally considered as the best fit model. Among the five models tested, logistic model, log linear model and linear models were found to be the best models for overall body weight, total length and head length respectively. For claw length and last segment length, log linear model was found to be the best model. These models can be used to predict growth rates in M. rosenbergii. However, further studies need to be conducted with more growth traits taken into consideration

Interactions of age-dependent mortality and selectivity functions in age-based stock assessment models

The natural mortality rate (M) of fish varies with size and age, although it is often assumed to be constant in stock assessments. Misspecification of M may bias important assessment quantities. We simulated fishery data, using an age-based population model, and then conducted stock assessments on the simulated data. Results were compared to known values. Misspecification of M had a negligible effect on the estimation of relative stock depletion; however, misspecification of M had a large effect on the estimation of parameters describing the stock recruitment relationship, age-specific selectivity, and catchability. If high M occurs in juvenile and old fish, but is misspecified in the assessment model, virgin biomass and catchability are often poorly estimated. In addition, stock recruitment relationships are often very difficult to estimate, and steepness values are commonly estimated at the upper bound (1.0) and overfishing limits tend to be biased low. Natural mortality can be estimated in assessment models if M is constant across ages or if selectivity is asymptotic. However if M is higher in old fish and selectivity is dome-shaped, M and the selectivity cannot both be adequately estimated because of strong interactions between M and selectivity.

Quantification of drag and lift imposed by pop-up satellite archival tags and estimation of the metabolic cost to cownose rays (Rhinoptera bonasus)

The recent development of the pop-up satellite archival tag (PSAT) has allowed the collection of information on a tagged animal, such as geolocation, pressure (depth), and ambient water temperature. The success of early studies, where PSATs were used on pelagic fishes, has spurred increasing interest in the use of these tags on a large variety of species and age groups. However, some species and age groups may not be suitable candidates for carrying a PSAT because of the relatively large size of the tag and the consequent energy cost to the study animal. We examined potential energetic costs to carrying a tag for the cownose ray (Rhinoptera bonasus). Two forces act on an animal tagged with a PSAT: lift from the PSATs buoyancy and drag as the tag is moved through the water column. In a freshwater flume, a spring scale measured the total force exerted by a PSAT at flume velocities from 0.00 to 0.60 m/s. By measuring the angle of deflection of the PSAT at each velocity, we separated total force into its constituent forces — lift and drag. The power required to carry a PSAT horizontally through the water was then calculated from the drag force and velocity. Using published metabolic rates, we calculated the power for a ray of a given size to swim at a specified velocity (i.e., its swimming power). For each velocity, the power required to carry a PSAT was compared to the swimming power expressed as a percentage, %TAX (Tag Altered eXertion). A %TAX greater than 5% was felt to be energetically significant. Our analysis indicated that a ray larger than 14.8 kg can carry a PSAT without exceeding this criterion. This method of estimating swimming power can be applied to other species and would allow a researcher to decide the suitability of a given study animal for tagging with a PSAT.

Demographic assessment of the blue crab (Callinectes sapidus) in Chesapeake Bay using extractable lipofuscins as age markers

The blue crab (Callinectes sapidus) plays an important economic and ecological role in estuaries and coastal habitats from the Gulf of Mexico to the east coast of North America, but demographic assessments are limited by length-based methods. We applied an alternative aging method using biochemical measures of metabolic byproducts (lipofuscins) sequestered in the neural tissue of eyestalks to examine population age structure. From Chesapeake Bay, subsamples of animals collected from the 1998–99 (n=769) and 1999–2000 (n=367) winter dredge surveys were collected and lipofuscin was measured. Modal analysis of the lipofuscin index provided separation into three modes, whereas carapace-width data collected among the same individuals showed two broad modes. Lipofuscin modal analysis indicated that most adults (carapace width >120 mm) were <2 years old. The results indicate that use of extractable lipofuscin can provide a more accurate and better resolved estimation of demographic structure of blue crab populations in the field than size alone.

Estimation of maximum sustainable yield (MSY) of hilsa (Tenualosa ilisha Ham.) in the Meghna river of Bangladesh

MSY per recruit of Tenualosa ilisha in the Meghna river was predicted as 112 g per recruit at the F(msy)=0.6/yr and at T(c)=0.6/yr. But Y/R=95 g per recruit was obtained at the existing fishing level, F=1.14/yr and at T(c)=0.6/yr. Existing F level was nearly double than the F(msy) level. Fishing pressure should be reduced immediately from F=1.14/yr to F(msy)=0.6/yr. F(msy)=1.14/yr was the same at first capture, T(c)=1.0, 1.2 and 1.4/yr, and MSY could be obtained as 142 g, 162 g and 176 g per recruit respectively. It is easier to change the first capture age (Tc) rather than changing off level. So, hilsa fishery manager may adopt F(msy)=1.14/yr while age at first capture must be increased from T(c)=0.6/yr (3 cm size group) to T(c)=1.4/yr (25 cm size group), by which 1.8 times production could be increased than the present production. MSY also possible to obtain as 201 g and 210 g per recruit at F(msy)=2.0/yr and 4.0/yr at T(c)=1.7/yr and 1.9/yr respectively. Under both the situations, hilsa production could be increased 2 times than the present production. To obtain the MSY=210 g per recruit the fishing level could be increased up to F=4.0/yr at T(c)=1.9/yr (34 cm size group). Economic point of view, hilsa fishery managers may choose to obtain the economic MSY as 201 g per recruit at F(msy)=2.0/yr and T(c)=1.7yr (31 cm size group) in the Meghna river of Bangladesh.