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 z0 is at a stable equilibrium when it occupies a fraction c0 of the sites. The per capita reproduction rate of an invading species m is
We know that
f(z0) = 1-c0-z0
= 0, which implies that
c0 = 1-z0. A necessary condition for this
species to be able to repel any invader is f'(z0)=0
c0 = 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(-z0) = b(z0). Because b is concave down for positive arguments, b(z0) < z0 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.