# Info

rowth rate k

Figure 2.4. Comparison of predicted (by Eq. (2.15)) and inferred age at maturity for different species of Tilapia, shown as an inset, and the 1:1 line. Data from Lorenzen (2000).

Figure 2.4. Comparison of predicted (by Eq. (2.15)) and inferred age at maturity for different species of Tilapia, shown as an inset, and the 1:1 line. Data from Lorenzen (2000).

0.5 1 1.5 2 ge at maturity predicted

the choice of m and follow one of the curves. The theory then predicts that as growth rate increases, age at maturity declines. If we fix growth rate and take a vertical slice along these three curves, the prediction is that age at maturity declines as mortality decreases. Each of these predictions should make intuitive sense and you should try to work them out for yourself if you are unclear about them. An example of the level of quantitative accuracy of this simple theory is given in Figure 2.4, in which I shown predicted (by Eq. (2.15)) and observed age at maturity for about a dozen species of Tilapia (data from Lorenzen 2000). Fish, like people, mature at different ages, so that when we discuss observed age at maturity, it is really a population concept and the general agreement among fishery scientists is that the age at maturity in a stock is the age at which half of the individuals are mature. Also shown in Figure 2.4 is the 1:1 line; if the theory and data agreed completely, all the points would be on this line. We see, in fact, that not only do the points fall off the line, but there is a slight bias in that when there is a deviation the observed age at maturity is more likely to be greater than the predicted value than less than the predicted value. Once again, we have the thorny issue of the meaning of deviation between a theoretical prediction and an observation (this problem will not go away, not in this book, and not in science). Here, I would offer the following points. First, the agreement, given the relative simplicity of the theory, is pretty remarkable. Second, what alternative theory do we have for predicting age at maturity? That is, if we consider that science consists of different hypotheses competing and arbitrated by the data (Hilborn and Mangel 1997) it makes little practical sense to reject an idea for poor performance when we have no alternative.

Note that both m and k appear in Eq. (2.15) and that there is no way to simplify it. Something remarkable happens, however, when we compute the length at maturity L(tm), as you should do now.

Show that size at maturity is given by

If you were slick on the way to Eq. (2.15), you actually discovered this before you computed the value of tm. This equation is remarkable, and the beginning of an enormous amount of evolutionary ecology and here is why. Notice that L(tm)/L1 is the relative size at maturity. Equations (2.15) and (2.16) tell us that although the optimal age at maturity depends upon k and m separately, the relative size at maturity only depends upon their ratio. This is an example of a life history invariant: regardless of the particular values of k and m for different stocks, if their ratio is the same, we predict the same relative size at maturity. This idea is due to the famous fishery scientist Ray Beverton (Figure 2.5) and has been rediscovered many times. Note too that since we conclude that if relative size at maturity for two species is the same, then since m/k will be the same (by Eq. (2.16)) that mtm must be the same.

All of our analysis until this point has been built on the underlying dynamics in Eq. (2.9), in which we assume that gain scales according to area, or according to W2/3. For many years, this actually created a problem because whenever experimental measurements were made, the scaling exponent was closer to 3/4 than 2/3. In a series of remarkable papers in the late 1990s, Jim Brown, Ric Charnoff, Brian Enquist, Geoff West, and other colleagues, showed how the 3/4 exponent could be derived by application of scaling laws and fractal analysis. Some representative papers are West et al. (1997), Enquist et al. (1999), and West et al. (2001). They show that it is possible to derive a growth model of the form dW/dt = aW3/4 — bW from first principles.

In the growth equation dW/dt = aW3/4 — bW, set W = Hn, where n is to be determined. Find the equation that H(t) satisfies. What value of n makes it especially simple to solve by putting it into a form similar to the von Bertalanffy equation for length? (See Connections for even more general growth and allometry models.)

Notes on the Use of Theoretical Models in the Study of the Dynamics of Exploited Fish Populations