## Growth Models and Backcalculation

Growth models are a standard product of length at age data. The models can vary in complexity from that of a simple straight line through length at age data (simple linear regression), to sophisticated maximum likelihood estimates of size at age. In most cases, the rationale for model preparation is to allow prediction of an expected mean size or growth rate at a given age, or to facilitate comparisons of estimated growth with other published estimates.

Calculations of growth rate may be based on equations derived from either empirically-fitted curves or one of the generally accepted growth models. There are many possible growth models, all of which can be applied to either length or weight data and use either daily or yearly ages. Frequently-used models include linear regression, Gompertz, von Bertalanffy, exponential and the logistic model. Equations for all of these, as well as others, are presented in Campana and Jones (1992). Also presented in Campana and Jones (1992) is the equation for a growth model which incorporates both age and temperature on a daily basis, thus allowing for changes in growth rate through time due to temperature changes.

Growth backcalculations used to estimate fish length at a previous age or date can be derived from a series of growth increments (either daily or yearly) and represent one of the most powerful applications of the otolith. Since the fish length:otolith length relationship can be determined, the widths of the daily (or yearly) growth increments in an otolith reflect the daily (or yearly) growth rates of the fish at that age and on those dates. Similarly, the radius of the otolith at a given age/increment is a reflection of the length of the fish at that age and on that date. If the fish length:otolith length relationship is linear, the increment widths are roughly proportional to the growth of the fish. Conversely, if the relationship is nonlinear, a more complicated conversion must be applied.

A major constraint to most existing backcalculation procedures is the assumption that the fish-otolith relationship is not only linear, but does not vary systematically with the growth rate of the fish. However, many studies have demonstrated that otoliths of slow-growing fish tend to be larger and heavier than those of fast-growing fish of the same size, whether at the daily or yearly scale. Such a systematic variation implies that growth backcalculations made with any of the traditional equations (eg- regression, Fraser-Lee or those of Francis (1990)) will tend to underestimate previous lengths at age, with the degree of error varying with the range of growth rates that are present in the population. The degree of error can be substantial in some cases, and appears to explain many reported cases of Lee's Phenomenon.

The presence of relatively large otoliths in slow-growing fish of a given species is a widespread phenomenon. To avoid backcalculation errors due to this effect, the biological intercept procedure uses a biologically-determined, rather than a statistically-determined, intercept in the backcalculation equation. Like the Fraser-Lee method, the biological intercept method assumes a linear relationship between fish length and otolith length within an individual fish. However, unlike the Fraser-Lee method, the value of the biological intercept is determined by the mean size of the fish and the otolith at the larval or juvenile stage, and thus is completely insensitive to any growth-related variations in the fish-otolith relationship. The equation for this method is:

where L_{a} is the backcalculated length of the fish at age a, L_{c} and O_{c} are the size of the fish and otolith at capture, respectively, and L_{i} and O_{i} are the size of the fish and otolith at the biological intercept, respectively.

What value should be used for the biological intercept? The biological intercept (fish length and otolith length) should be measured in the smallest fish possible AS LONG AS all subsequent fish and otolith growth is linear (proportional). It is VERY IMPORTANT that very young fish with a nonlinear fish-otolith growth trajectory not be used, since the resulting backcalculations will not be as accurate as they could be. Therefore, the biological intercept of some species may be at the juvenile stage, while others may be right at the time of hatch.

The biological intercept will always yield backcalculated values which are at least as accurate as those of the regression or Fraser-Lee methods. Therefore, there is no disadvantage to the use of the biological intercept method, other than those that are shared by the other proportional methods. To quickly determine if the Fraser-Lee method will yield comparable results to that of the biological intercept method, simply compare the value of the biological intercept with the predicted fish length derived from the population fish-otolith regression for a comparable otolith length: if they are significantly different, significant gains in accuracy can be expected by using the biological intercept method.

For further information on growth models, growth backcalculation and the biological intercept method, see Campana and Jones (1992) and Campana (1990).