Assignment 4 Solution

Description

Supply networks and allometry:

1. (3 + 3 points)

This question’s calculation is a specific, exactly-solvable case of the general result that you’ll will attack (with optional relish and other condiments) in the following question.

Consider a set of rectangular areas with side lengths L₁ and L₂ such that

L₁ / A 1 and L₂ / A 2 where A is area and ₁ + ₂ = 1. Assume ₁ > ₂ and that ϵ = 0.

Now imagine that material has to be distributed from a central source in each of these areas to sinks distributed with density (A), and that these sinks draw the same amount of material per unit time independent of L₁ and L₂.

1. Find an exact form for how the volume of the most efficient distribution network scales with overall area A = L₁L₂. (Hint: you will have to set up a double integration over the rectangle.)

2. If network volume must remain a constant fraction of overall area, determine the maximal scaling of sink density with A.

Extra hints:

• Integrate over triangles as follows.

• You need to only perform calculations for one triangle.

	x2			= L2 x
		x	2		1
				L1
	L2
	2
			dx1dx2
2	2
				1
1
L1	L	1		x1
− 2	2
1	1
2	2
	L2
	− 2

3. (a) For a family of d-dimensional regions, with scaling as per the previous question, determine, to leading order, the scaling of hyper-surface area S with volume V . In other words, find the exponent in S / V as V ! 1.

Assume that nothing peculiar happens with the shapes (as we have always implicitly done), in that there is no fractal roughening.

Hint: As a start, figure out how the circumference for the rectangles in question 1 scales with area A. For d dimensions, think about how the hyper-surface area of a hyperrectangle (or orthotope) would scale.

2. For general d, what is the minimum and maximum possible values of and for what values of the _i does these extrema occur?

The goal and a connection to energy metabolism:

3