###
*Ecological Archives* E081-027-A2

**Frederick R. Adler, and Julio Mosquera. 2000. Is space necessary?
Interference competition and limits to biodiversity. ***Ecology* 81:3226-3232.

**Appendix B: **Analytic functions support only discrete species
To prove that an analytic function *b* cannot support a stable coalition
with a continuum of species, we require the following two lemmas. First, we
assume that the coalition *p*(*m*) has ``compact support", meaning
that the mortality rates are bounded above, and that it has a finite integral
(as it must). Then *p*(*m*) is an *L*^{1} function. The
key result is that if *b* is an analytic function, the convolution of *p*
with *b* will also be analytic, which follows from Morera's Theorem (Rudin,
1974).

Given this lemma, we have that *f*(*m*) is analytic if *b* is.
The same must hold for the second derivative of *f*. If the coalition included
a continuum of species, the second derivative would have to be zero on an interval.
However, an analytic function cannot be exactly 0 on an interval without itself
being identically zero everywhere. This is only possible if the second derivative
of *b* is zero, or if *b*(*z*) is a linear function. Such a competitiveness
function cannot be bounded above. Furthermore, solutions with this competitiveness
function are degenerate, supporting an infinite number of neutrally stable coalitions
(results not shown).

*Literature Cited*

Rudin, W. 1974. Real and Complex Analysis. McGraw-Hill, Inc., New York, New
York, USA.

[Back to E081-027]