Multiple birth states still necessitate the calculation of an eigenvalue. Luckily, for a number of calculations it is possible to use as fitness proxy the explicitly calculable quantity
In particular, provided the set X of potential trait vectors is connected, X*is a global ESS if q(X|X*) < 0 for all X = X*, and only if q(X|X*) < 0 for all X. (A strategy is called a local ESS when it cannot be invaded by nearby mutants and a global ESS when it cannot be invaded by any mutant.) Moreover, any local ESSs can be calculated and their attractiveness determined by substituting q for invasion fitness.
Only when it comes to the calculation of evolutionary transients there is a difference. For small mutational steps r (Xm |Xr) « (Xr |Xr)(Xm - Xr)
with progressively more complicated corrections appearing in the higher-order terms.
Similar results hold good for models described in terms of  or , with Tb = 1 and in the case of , q(Xm|Xr) = P(Xm|Xr)(0) and
Matrix models with T-periodic environments can be brought into the fold by replacing A in  with C(T), as defined in the section on calculation procedures, and replacing Tb with T in  .
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