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

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

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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