### Jetz, Carbone, Fulford and Brown: The Scaling of Animal Space Use

I want to transition to more ecological content here, which really started with my post on extinctions and climate change. On to *Science* where some folks, including Jim Brown, are talking about how space use changes as body size changes.

Not surprisingly, larger animals need more space. In general, area scales with the square (exponent=2), length scales linearly (exponent=1), and volume (body size, mass, etc.) scales with the cube (exponent=3). So you expect that an organism that gets 2 centimeters longer will also increase its surface area by 4 square centimeters, and add 8 cubic centimeters of volume (8 milliliters, or 8 grams of water). An increase of surface area of 16 square centimeters would translate to 16^{2/3} cubic centimeters (that's the 2/3 power), or 64 cc. That's the magic of scaling.

A lot of recent work in Brown's lab has focussed on energy as it scales with body size, and found a consistent relationship that energy use - metabolic rate - (B) is proportional to M^{3/4} where M is body mass and 3/4 is an exponentiation. The cubic power makes sense, but the quarter power is the magic of energy scaling. That the power is less than 1 means that there are economies of scale; as you get bigger, you use less energy per unit of mass, although you need more net energy to live. The need for more energy with larger bodies implies that you need more foraging area as you get larger. In general, you expect that home range would have some close interaction with the range of individual organisms.

To derive the actual form of the relationship, you have to integrate ecological interactions. We could reasonably expect that home range area would scale with the 2/3 power of body mass, since we're going from a cubic variable to a squared variable. Various papers have empirically found that body mass scales linearly with home range, which is somewhat surprising. To understand, we have to integrate some ecology.

There are two basic mathematical ways to get to linear scaling.

- Resource availability decreases with body size at the 1/4 power, while the proportion of the available resources that the owner of a home range has exclusive access to does not scale at all (exponent = 0); or
- Resource availability does not scale and the owner's exclusive access proportion scales with a -1/4 power.

Macroecological evidence shows no scaling in resource availability with body size, so they proceed to unwind the ecology of exclusive access scaling.

Various people, including my advisor, have argued that home range scaling ought to have some relationship with overlap between neighboring ranges. It makes a lot of sense that a larger area is harder to defend from neighbors, so the fraction that is exclusively exploited by one individual will decline with body size. (Again, movement is one dimensional, area is 2D. If you can move twice as fast, that means you can move through a fixed distance faster, but twice the space takes 4 times longer to search). Since home range size scales with the 3/4 power of body mass, and we see the area being used in a day scaling with 1/4 power, that means home range size gets very much larger than the area used in a day very fast.

Work through the math, and you get home range proportional to mass / resource density. The math is interesting, but fairly a straightforward application of the theory of random walks and some results from statistical physics.

By working through some regression analyses, Jetz, et al. show that there is no significant relationship between resource density and mass, justifying an important assumption. They also could separate the exclusive portion of the home range from total home range size.

This is the interesting part, because they show that at very low masses (where most mammals are), mammals use their entire home range exclusively, but that at 1 kg, the owner only has exclusive access to 1/3 of the resources, and at 100 kg, only 7%.

Brown and some collaborator's have demonstrated that 100g seems to be some sort of optimal body size for mammals, and it wouldn't be surprising if that was also a place where the divergence between home range size and exclusively used space gets noticeable.

The image above is from the accompanying commentary by Steven Buskirk, who points out that this line of analysis has powerful possibilities. A lot of work by Brown and his colleagues has shown surprisingly widespread connections between ecological, physiological and evolutionary processes, all linked by that 1/4 exponent, which means that energy - or metabolic rate - is the thread that could unify ecology and organismal biology. There's still a lot of basic research to be done on those questions, but the possibilities are exciting.

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