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, We now seek to decompose equation (Eq. S14) in stabilizing and equalizing terms as in 1226

&. Chesson, To find the stabilizing term, we first looked at the long-term 1227 growth rate at low density when all the species have the same fitness, 2000.

, We now search for the corresponding equalizing term in an additive decomposition as in 1231

&. Chesson, Starting from (Eq. S14), we obtained the 1232 following final expression for the long-term growth rate of a species, framework (Eq. 1 above), p.1233, 2000.

, We thus retrieve in (Eq. S16) a stabilizing term with a form consistent with Chesson, 2000.

, Note that changing affects but not , which is a desirable property of the 1239 equalizing/stabilizing decomposition. The main difference with Chesson's (2000) framework 1240 is that the is not a simple arithmetic mean of the fitnesses but also has to include some 1241 interaction effects of other species

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