Fitting growth models to length frequency data

Geoff M. Laslett 0 J. Paige Eveson 0 Tom Polacheck 0 0 G. M. Laslett: CSIRO Mathematical and Information Sciences , Private Bag 10, Clayton South MDC, Clayton, Victoria 3169 , Australia. J. P. Eveson and T. Polacheck: CSIRO Marine Research, GPO Box 1538, Hobart, Tasmania 7001 , Australia A novel two-stage procedure for fitting growth curves to length frequency data collected from commercial fisheries is described. The method is suitable for species in which cohorts are spawned over a limited time period, and samples of length frequency data are collected regularly (e.g. in weekly, fortnightly, or monthly time intervals) over an extended time period. In the first stage of analysis, Gaussian mixtures are fitted separately to the data for each time interval, and summary statistics (component means and standard errors) are extracted. In the second stage, parametric growth models, such as the von Bertalanffy seasonal growth curve, are fitted to the summary data. The error structure in this second stage of analysis incorporates random between-year effects, random within-year age-group effects, random within-year time-interval effects, random within-year age-group and timeinterval interactions, and sampling errors. This complex error structure incorporating unbalanced crossed and nested random effects acknowledges that commercial fishing is not an exercise in random sampling, and allows for the inevitable additional sources of random variation in such an enterprise. The method is applied to South Australian southern bluefin tuna length frequency data collected from 1964 to 1989, and leads to the conclusion that juvenile tuna grew faster in the 1980s than in the 1960s, with the 1970s being a decade of highly variable growth. Introduction Valuable information about the growth of fish can often be extracted from length data that have been collected regularly over an extended time period. Such data often exist for commercially harvested species where routine length sampling of the catch occurs. If a species has a restricted spawning period, then fish belonging to the same cohort and caught around the same time will exhibit a limited range of lengths. For young fish, which are growing quickly, the overlap in the length ranges between ages is often small enough so that the length frequency distribution will show obvious modes. For older fish, the overlap in length ranges becomes progressively greater so that the modes become more difficult to distinguish. The progression of modal lengths over time can be tracked to give information on the growth of young fish. Length frequency data provide information on two aspects of growth. First, yearly growth can be estimated by comparing the average length of one-year-olds, twoyear-olds, three-year-olds, and so on caught at the same time. Second, seasonal growth can be inferred by following the growth of a particular age group within a year. Other data sources, such as tagerecapture surveys and direct ageing data from hard parts analyses, often do not exist on a regular enough time scale to be able to provide detailed information on seasonal growth. Length frequency data are important for this reason. Extracting the information on growth from length frequency data is not straightforward. First, length frequency data do not come with any independent age attribution so the researcher has to assign the fish to age classes, either explicitly or statistically. Second, the spawning period for a species may be several months and there may be peaks in spawning activity within this period. Such variable spawning can complicate the modal decomposition, and also the growth analysis because growth patterns of fish that were spawned early in the season may differ from those spawned later. Third, length data are often collected from commercial fisheries. In one sense, fisheries data are more informative than data from scientific research programs because they are more abundant and more consistent over time. However, fishing is not designed as a random sampling exercise and, consequently, caution must be used in treating the length data as an unbiased random sample of the population. Finally, measurement error is endemic and may be dependent on the measurer. It is important to develop methods of data analysis that capture these sources of variation. This article presents our method for extracting growth information from length frequency data. Our method has some features in common with other methods presented in the quantitative fisheries literature (Fournier et al., 1990; Leigh and Hearn, 2000), but departs from them in significant ways. In particular, we develop a two-stage analysis. In the first stage, each length frequency distribution is decomposed into age groups using a Gaussian mixture model and relevant summary statistics are extracted. In the second stage, the summary statistics are used as raw data for growth modelling. This approach allows us to explore and visualize the sources of variation in the data prior to final modelling. More direct (i.e. single-stage) methods are likely to overlook the many possible complications in real length frequency data. We illustrate our method on southern bluefin tuna length data collected from the South Australian surface fishery. The surface fishery operates annually from approximately November to July of the following year, and is the largest fishery in Australia. The catches are sampled for length regularly throughout each fishing season and aggregated half-monthly, so a consistent and long-term time-series of length data exists. Additionally, the surface catches consist predominantly of juvenile fish aged 1e5, and therefore are ideal for modal length analysis. In this article, we first discuss the form of the southern bluefin tuna length frequency data used in our analysis and some of its features. We outline our method of analysis, in which we fit a growth model to summary statistics derived from fitting mixture models to each length frequency sample. Finally, we apply the method to southern bluefin data and discuss the results and the method in general. The data We focus on length frequency data collected from southern bluefin tuna caught in the South Australian surface fishery from 1964 through 1989. During this time, fish were caught for the canning market using purse-seine vessels, which use nets that catch the majority of fish in a school and do not have the ability to select by size within a school. In the 1990s, the focus of the fishery switched to catching tuna for the sashimi market using pole and line vessels, which target larger fish within schools. In order to avoid biases from size-selective fishing, we only include data prior to 1990 in our analysis. The length data were collected using a two-phase sampling procedure. First, within each half-month, landings of tuna were selected for detailed study. Second, a sample of tuna was taken from each selected landing and the fork length (...truncated)