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

**Appendix C**: An uninvadible single species must have *m*=0

Suppose that a single species *z*_{0} is at a stable equilibrium
when it occupies a fraction *c*_{0} of the sites. The per capita
reproduction rate of an invading species *m* is

We know that
*f*(*z*_{0}) = 1-*c*_{0}-*z*_{0}
= 0, which implies that
*c*_{0} = 1-*z*_{0}. A necessary condition for this
species to be able to repel any invader is *f*'(*z*_{0})=0
or

so that
*c*_{0} = 1/*b*'(0). But we can now compute the invasion rate
of the species with *m*=0 as

where we used the fact that
*b*(-*z*_{0}) = *b*(*z*_{0}). Because *b*
is concave down for positive arguments,
*b*(*z*_{0}) < *z*_{0} *b*'(0), which
implies that *f*(0) > 0. Thus, any single species with *m* >
0 can be invaded by *m*=0. The only possible uninvadible single species
has *m*=0.