Frederick R. Adler, and Julio Mosquera. 2000. Is space necessary? Interference competition and limits to biodiversity. Ecology 81:3226-3232.
Suppose the coalition includes all species between m=0 and m=zmax.
For stability, we require that
for . Differentiating with respect to mgives that 2s p(m)=1 or for . Therefore, .
where the integral is taken over all members of the coalition,
because f(m)=0 on this set. However,
where is the mean of m. The interaction terms cancel because losses by the species with strategy z are gains by the species with strategy m (as can be shown by changing the order of integration in the second term).
We can then solve for . With the additional relation for a uniform distribution, we find . This coalition will be stable to invasion because any species with m > zmax has no competitive advantage over zmax itself, but at the cost of higher mortality. In terms of the invasion function f, f(m) < f(zmax) = 0.