Estimating reaction norms for age and size at maturation with reconstructed immature size distributions: a new technique illustrated by application to Northeast Arctic cod

Mikko Heino Ulf Dieckmann Olav Rune God Reaction norms for age and size at maturation describe the probability of immature fish maturing at a certain age and size. Knowledge of such reaction norms is increasingly important, both for observing and understanding changes in maturation and for calibrating size- and age-structured population models. Estimating the reaction norms for age and size at maturation by logistic regression requires data on the size and age distribution of immature and maturing fish. To permit such estimation when measurements of the size and age distribution of immature fish are not available, the information can be reconstructed by means of a back-projection procedure. For the reconstruction, only the size and age distribution of maturing fish, or first-time spawners, together with the age-dependent proportion of mature fish, given in the form of maturity ogives, have to be known. The method of Gulland (1964) is used to generate maturity ogives in the absence of data on immature fish. The robustness of the estimation method is demonstrated by analysing artificial data generated with a known reaction norm. Application of the reconstruction method to a set of measurements on Northeast Arctic cod (Gadus morhua) collected between 1933 and 1944 illustrates how the approach allows new information to be extracted from real data. 1054-3139/02/060562+14 $35.00/0 - Age and size at maturation are among the key life history traits that facilitate the adaptation of fish to environmental variability (Roff, 1992; Stearns, 1992). At an individual level, age and size at maturation influence growth rate, fecundity, and survival probability later in life. At a population level these effects interact, such that changes in maturation properties affect age and size distributions, population dynamics, and productivity. In particular, changes in maturation properties influence the potential yield from fish stocks. In the modern era of the precautionary approach, observing and understanding changes in the maturation properties of exploited stocks is critical to fisheries management. Within each stock, individual fish typically follow different growth trajectories, maturity being attained by different individuals at different combinations of age and size (Alm, 1959; Stearns and Crandall, 1984). Moreover, because individual maturation is influenced also by the condition of the fish (Bernardo, 1993), involving factors such as recent growth history and the quantity of body reserves, it is common to find that, even within a given age- and size-class, some fish have matured while others have not. The process of maturation therefore calls for a probabilistic treatment. A convenient way to characterize the dependence of maturation on age and size is the so-called probabilistic reaction norm for age and size at maturation (maturation reaction norm, for short). This reaction norm describes the age- and sizedependent likelihood that an immature fish matures during the season considered (Heino et al., 2002). The maturation reaction norm, estimation of which is equally valuable from both theoretical and applied perspectives, is introduced in the following section. As explained below, estimating the maturation reaction norm requires information on both immature and maturing fish. Unfortunately, however, situations in which only data on mature (or maturing) fish can easily be collected are not uncommon in stock assessment: juveniles may have an inconspicuous lifestyle, or they may be spatially segregated from adults. The latter applies to species with a spawning migration, such as Northeast Arctic cod. For such species it would, at first glance, seem that estimating maturation reaction norms is precluded; fortunately this is not the case. In this paper we introduce a new technique by which such missing data can be reconstructed through a backprojection procedure. This procedure utilizes information on the size distribution of maturing fish, on age-based maturity ogives, and on growth curves of immature and maturing individuals. We show that the reconstruction method permits the robust estimation of probabilistic maturation reaction norms from stock data that lack information on immature size distributions. The method aims at estimating the size distributions of juveniles when these have not been measured directly. However, it does require information on juveniles in the form of age-based maturity ogives. This apparent contradiction has two solutions. First, estimating age-based maturity ogives only requires information on relative juvenile abundance, not on their size distribution; consequently, age-based maturity ogives may well be known even if the size distributions of juveniles have not been measured. Second, if juvenile data are completely lacking, Gullands (1964) method still allows estimation of age-based maturity ogives. [Gulland (1964) originally devised this method for Northeast Arctic cod; for subsequent applications, see Jrgensen (1990), Ajiad and Jakobsen (2001), and the section Robustness of the reconstruction technique later.] The method is illustrated by estimating reaction norms for data on Northeast Arctic cod (Gadus morhua L.), commercially one of the most important gadoid stocks in the world. Mature Northeast Arctic cod undertake an annual spawning migration from the Barents Sea to the northern Norwegian coast, the main spawning locations being located in waters off the Lofoten Islands (Bergstad et al., 1987). Representative samples of both immature and maturing cod are available from 1984 onwards when annual winter surveys were started. However, records of age and length of spawning cod around the Lofoten Islands go back as far as 1932. The goal of analysing temporal changes in age and size at maturation in this exceptionally long time-series prompted us to develop a method that would allow extracting more information from the available data. We demonstrate the estimation of probabilistic reaction norms and the underlying back-projection procedure by reconstructing the reaction norms for age and size at maturation for the Northeast Arctic cod cohorts of 19261931, which matured over the years 19331944. Reaction norms for age and size at maturation In general, reaction norms describe how one genotype can give rise to distinct phenotypes when exposed to different environmental conditions (Schmalhausen, 1949). In particular, the reaction norm for age and size at maturation describes how variability in growth conditions, reflected by variations in size-at-age, influences maturation. The maturation reaction norm is defined here as the probability that immature fish mature during a given time interval and at a certain age and size (Figure 1; Heino et al., 2002). A description of the entire reaction norm involves specifying these probabilities for all relevant ages and sizes. This probabilistic definition is a generalization of the essentially deterministic definit (...truncated)