Making It Fit by Leaf and Murphy

Part II – The von Bertalanffy Growth Function

Dr. Latimer looked through the textbooks that he had and found that:

“Theoretical nonlinear curves are often used to describe the relation of age to length in fishes. One such curve that assumes indeterminate, asymptotic growth and is often used to describe length-at-age is the von Bertalanffy growth function.”

The von Bertalanffy growth function (VBGF) can be expressed in the following form:

Karl Ludwig von Bertalanffy was an Austrian born biologist and intellectual who published the above model in 1938. This model (presented above in its two-parameter form) is perhaps one of the most widely used models to describe age and growth relationships of fishes. The above equation describes how an individual’s (the i ^th individual’s) length, L, is predicted as a function of its age (t) as a result of the values of the parameters k and L_∞. k is the instantaneous growth rate (units of time, most often an annual period, year^-1) and L_∞ is the maximum length (units are length, such as inches, mm, or cm) attained by an average individual in the population. Below we present the age as the annuli count, which are often assumed to be annual.

This curve looks like:

In this model, the slope of the line defined by the VBGF changes (decreases) as age increases. At some age (around age 8), the individual stops growing.

Although it had been a while since Dr. Latimer had done this kind of work, he thought that he could make some headway if he compared it to a similar problem—how to fit a straight line to data. A straight line that describes the relationship of data is called a linear model and is described mathematically with two parameters, m and b. m describes the slope of the line and b is the location that the line crosses the y axis.

y_i = mx_i + b + ε_i

The ε_i term describes the amount of error, or residual error, that is not accounted for by the prediction (the difference between the observed length, y_i, and the expected length, mx_i + b, predicted by the model). He knew that in order to fit a line, the value of the difference of the observed data (that collected in the field) and expected data (the line) needed to be as small as possible. This is called “minimizing the sum of squared residuals.” Thus the value of ε for each individual is the residual.

Linear model: y_i = mx_i + b + ε_i

von Bertalanffy growth function: L_i = L_∞ (1 − e^−kti) + ε_i

The observed data are on the left side of the equal sign, and the equation for the line is specified on the right side of the equation. The shapes of the non-linear or linear models are determined by the parameters.

Questions

1. Discuss the difference between “data” and “parameters.” What are parameters and why are they important for our understanding of what a line looks like?

   Figure 6

   

2. Why would we prefer the dashed line if we were trying to describe the trends in the data? What exactly are we describing with the dashed line?

3. Which line (solid or dashed) has a greater slope (m) and which has the greater y-intercept (b)?

4. Using estimates of the parameters, m and b, for the linear model, what is the equation for the dashed line (L_t is the length at age = t and age is t)?

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