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age, which we denote by W(t) and assume that mass and length are related according to W(t) = pL(t)3, where p is the density of the organism and the cubic relationship is important (as you will see). How valid is this assumption (i.e. of a spherical or cubical organism)? Well, there are lots of organisms that approximately fit this description if you are willing to forgo a terrestrial, mammalian bias. But bear with the analysis even if you cannot forgo this bias (and also see the nice books by John Harte (1988, 2001) for therapy).

The rate of change of mass is a balance of anabolic and catabolic factors dW

—— = anabolic factors — catabolic factors (2.8)

We assume that the anabolic factors scale according to surface area, because what an organism encounters in the world will depend roughly on the area in contact with the world. Thus anabolic factors = a—2, where a is the appropriate scaling parameter. Let us just take a minute and think about the units of a. Here is one example (if you don't like my choice of units, pick your own): mass has units of kg, time has units of days, so that dW/dt has units of kg/day. Length has units of cm, so that a must have units of kg/day • cm2.

We also assume that catabolic factors are due to metabolism, which depends on volume, which is related to mass. Thus catabolic factors = cL3 and I will let you determine the units of c. Combining these we have dW = aL2 - cL3 (2.9)

Equation (2.9) is pretty useless because W appears on the left hand side but L appears on the right hand side. However, since we have the allometric relationship W(t) = pL(t)3

and if we use this equation in Eq. (2.9), we see that

so that now if we divide through by 3pL2, we obtain dL a c

dt 3p 3p and we are now ready to combine parameters.

There are at least two ways of combining parameters here, one of which I like more than the other, which is more common. In the first, we set q = a/3p and k = c/3p, so that Eq. (2.12) simplifies to dL/dt = q — kL. This formulation separates the parameters characterizing costs and those characterizing gains. An alternative is to factor c/3p from the right hand side of Eq. (2.12), define L1 = a/c, which we will call asymptotic size, and obtain

This is the second form of the von Bertalanffy growth equation. Note that asymptotic size involves a combination of the parameters characterizing cost and growth.

Check that the units of q, k and asymptotic size are correct.

Equation (2.13) is a first order linear differential equation. It requires one constant of integration for a unique solution and this we obtain by setting initial size L(0) = L0. The solution can be found by at least two methods learned in introductory calculus: the method of the integrating factor or the method of separation of variables.

Show that the solution of Eq. (2.13) with L(0) = L0 is

In the literature you will sometimes find a different way of capturing the initial condition, which is done by writing Eq. (2.14) in terms of a new parameter t0: L(t) = LM(1 — e—k(t—10)). It is important to know that these formulations are equivalent. In Figure 2.2a, I show a sample growth curve.

For many organisms, initial size is so small relative to asymptotic size that we can simply ignore initial size in our manipulations of the equations. We will do that here because it makes the analysis much

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