Ecological Archives E081-027-A3

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

f(m) = 1 - c0 - m + c0 b(m-z0) .

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 or

f'(z0) = -1 + c0 b'(0) = 0

so that c0 = 1/b'(0). But we can now compute the invasion rate of the species with m=0 as

\begin{displaymath}f(0) = 1-c_0 + c_0 b(-z_0) = z_0 - \frac{b(z_0)}{b'(0)} ,

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.

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