We've talked some about scaling relationships and why they emerge but haven't gotten too much into the details of the theories attempting to explain why certain scaling properties exist. These fall into two camps, life history models that usually take certain extrinsic properties as givens and more complicated physical models that attempt to explain why metabolic rate is body mass to the 3/4 power from first principles of energetics and geometry. Examples of life history models predicting the allometries for traits like birth rate, mortality rate, age at first reproduction, and life span are those of Charnov (1991, 1993, 2001). These tend to take factors like the production function (growth rate is some constant a*mass^(3/4)) as a given and predict the other allometries, which tend to be +1/4 powers for times (life span, generation length, etc) or -1/4 powers for rates (birth rate, mortality rate, intrinsic rate of increase r, etc.). These models often have mortality rate as an external environmental parameter, but not always, and often take size at independence as a given (which is a linear function of adult mass and this predicts the -1/4 power scaling of fertility rate). One very successful geometric model is presented in West et al. (1997, 1999) and demonstrates that the 3/4 scaling of metabolic rate results from an optimal solution to the problem of efficiently constructing biological resource distribution networks that must deliver resources to all the cells in an organism while satisfying certain design characteristics. That is, the network should efficiently fill space and deliver resources to cells as effectively as possible. This problem generates a fractal distribution network that optimally fills space and generates a predicted 3/4 power of metabolic rate with mass. The 'networks' we are referring to in this context are vascular systems in plants and circulatory systems in animals. (This treatment is shockingly rudimentary but hopefully good enough for present purposes).
This paper by Lev Ginzburg and John Damuth takes a different view on scaling relationships in ecology by looking at the dimensionality of organisms. First, here's the citation info and abstract. I'll continue to comnent below.
The Space‐Lifetime Hypothesis: Viewing Organisms in Four Dimensions, Literally
1. Department of Ecology and Evolution, Stony Brook University, Stony Brook, New York 11794;
2. Department of Ecology, Evolution and Marine Biology, University of California, Santa Barbara, California 93106
Abstract:
Much of the debate about alternative scaling exponents may result from unawareness of the dimensionality appropriate for different data and questions; in some cases, analysis has to include a fourth temporal dimension, and in others, it does not. Proportional scaling simultaneously applied to an organism and its generation time, treating the latter as a natural fourth dimension, produces a simple explanation for the 3/4 power in large‐scale interspecies comparisons. Analysis of data sets of reduced dimensionality (e.g., data sets constructed such that one or more of the four dimensions are fixed), results in predictably lower metabolic exponents of 2/3 and 1/2 under one and two constraints, respectively. Our space‐lifetime view offers a predictive framework that may be useful in developing a more complete mechanistic theory of metabolic scaling.
The authors argue that organisms can literally be viewed as four dimensional objects, three spatial and one temporal. While many traits scale with body size, they specifically focus on the well-known finding that metabolic rate scales as the +3/4 power of body mass whereas lifespan goes as the +1/4 power. This makes the product of the two an isometric relationship (m3/4 x m1/4 = m1), such that a doubling in an organism’s size predicts a doubling in the energy it metabolizes in a lifetime. While many researchers take this as a consequence of other scaling relationships, it plays a fundamental role in the 4D view. As they state it, “these observations suggest instead that the scaling of lifetimes may reflect a fundamental manner in which organisms of all body masses are ecologically and evolutionarily functionally similar.” Thus the organism’s four dimensions, three spatial (length, area, volume) and one temporal (generation time) together give m1. If these four dimensions are evenly divided into the isometric scaling of lifetime metabolic rate then each will be m1/4. This predicts that metabolic rate should be m3/4 because energy is taken in through a 3D surface and then allocated to processes that take place in 4D (the dimension of time and within the 3 dimensional space of the organism). And if metabolic rate is m3/4, the remaining dimension, generation time (or lifespan), should be m1/4 to preserve the isometric scaling lifetime metabolic rate.
The role of generation time in ecology and evolution itself is another key component of the 4D argument: “Constructing one viable and reproductively capable daughter requires a certain duration (a “generation time”) that is conveniently viewed as an organism’s fourth dimension. So, on average, it takes a generation time of metabolism for a mother to guarantee the existence of her replacement.” This establishes the reasoning for why generation time is fundamentally an organism’s fourth dimension.
Where this argument becomes even less conventional is in the stated lack of a mechanism. In fact, my reading of the paper is that they intend the argument to predict the set of criteria to which any proposed mechanistic explanation of ¾ power scaling in biology must conform. For instance, they can predict that progressive reductions in dimensionality, by holding constant generation time, length, etc. should lead to predictable reductions in the exponent. So if generation time is held constant then they predict that metabolic rate should be a 2/3 power of mass, rather than ¾, and cite examples where within species metabolic rates have been shown to go as the 2/3 power of mass (if length and generation time are held constant, as with species of same size and lifetime, the scaling should be ½, etc.). They do this with multiple regressions. For instance, they predict that if height and generation length are controlled for, then metabolic rate should scale as the 1/2 power of mass and their data seem to conform to this prediction.
Because they do not suggest a mechanism, they are not necessarily at odds with any particular theory of metabolic scaling, such as that of the space-filling fractal geometry of supply networks in the circulatory and vascular systems of mammals and plants (e.g., West et al. 1999 - mentioned above). The explicitly non-mechanistic argument in the paper adds to its uniqueness but is also where some people may have the greatest trouble with the paper, as we are taught to focus on mechanisms and this nature of dimensional thinking is much more foreign to us (and maybe difficult to interpret at first). The theory makes simple and elegant predictions that should lead readily to either coherence or conflict with some of the existing takes on the topic (note that I'm saying the predictions are simple and elegant but am not saying anything about whether the empirical results are broadly accurate. Its of course too soon to see if how these predictions will weather the tests of time. They do give some good empirical support in the paper). Either way, dimensional thinking is a novel approach in this area that, when combined with the argument for the importance of generation time, makes a fundamental contribution to the literature and will certainly alter future approaches to the subject of scaling in ecology.
Well, something to think about anyway.
Oskar
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