Otolith growth models
Learn about growth rate calculations, backcalculations, constraints and the biological intercept method.
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Growth rate calculations
Growth models are a standard product of length at age data. The models can vary in complexity, including:
- sophisticated maximum likelihood estimates of size at age
- a simple straight line through length at age data (simple linear regression)
In most cases, the rationale for model preparation is to:
- allow prediction of an expected mean size or growth rate at a given age
- 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:
- Gompertz
- von Bertalanffy
- linear regression
- exponential and the logistic model
Another method is the equation for a growth model which incorporates both age and temperature on a daily basis. This allows for changes in growth rate through time due to temperature changes.
Backcalculations
We can use growth backcalculations to estimate fish length at a previous age or date derived from a series of growth increments (either daily or yearly). This is one of the most powerful applications of the otolith.
We can determine the fish length to otolith length relationship. Therefore, 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
- 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 to otolith length relationship is linear, the increment widths are roughly proportional to the growth of the fish. If it’s nonlinear, a more complicated conversion must be applied.
Constraints
Backcalculation procedures are constrained by the assumption that the fish-otolith relationship isn’t only linear, but that it doesn’t vary systematically with the growth rate of the fish.
However, 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. This systematic variation implies that growth backcalculations made with traditional equations (like regression, Fraser-Lee or Francis 1990) will tend to underestimate previous lengths at age. The degree of error will vary 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.
Biological intercept method
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. Thus, the intercept is completely insensitive to any growth-related variations in the fish-otolith relationship.
Biological intercept equation
The equation for this method is:
La=Lc+(O−Oc)(Lc−Li)(Oc−Oi)−1La=Lc+(O-Oc)(Lc-Li)(Oc-Oi)-1
Where:
- La is the backcalculated length of the fish at age ‘a’
- Lc and Oc are the size of the fish and otolith at capture, respectively
- Li and Oi are the size of the fish and otolith at the biological intercept, respectively
Biological intercept value
If all subsequent fish and otolith growth is linear (proportional), the biological intercept (fish length and otolith length) should be measured in the smallest fish possible.
It’s important that very young fish with a nonlinear fish-otolith growth trajectory not be used. Their resulting backcalculations won’t 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.
Advantages
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.
You can easily determine if the Fraser-Lee method will yield comparable results to that of the biological intercept method. To do so, 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’re significantly different, significant gains in accuracy can be expected by using the biological intercept method.
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