Theorem 1. As alpha approaches 0,
The function h(r) = log N(r) / 2log L corresponds to plotting
species number versus area on log-log paper. Theorem 1 says that if we do this
for alpha close to 0, then the result will be close to a curve
that is identically 0 out to r = 1 and then has slope 1 after that.
Theorem 2. If r > =1, then as alpha approaches 0,
It is, at first sight, disappointing that our limiting curve
given in Theorem 1 consists of two line segments with slopes 0 and 1,
rather than a single line with an "interesting" slope.
However, if we assume that the species area curve is approximately
linear on [0,1] we can compute the slope from the change in value
between r = 1 and r = 0
Taking alpha=10^{-k}, [L=1/sqrt(alpha)] and letting
k = 4, 8, 12 gives slopes of .331, .241, .190,
a range of
values comparable to that observed in the data. We are unfortunately
not able to prove that the species area curve is linear over the
indicated range. However, the following three simulated species-area
curves for the process on a 1000 x 1000 grid with alpha = 10^{-6}
supports this conclusion.
