## Still Simpler Proxy

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

with

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)

X1m 1

with progressively more complicated corrections appearing in the higher-order terms.

Similar results hold good for models described in terms of [1] or [2], with Tb = 1 and in the case of [2], q(Xm|Xr) = P(Xm|Xr)(0) and

Matrix models with T-periodic environments can be brought into the fold by replacing A in [23] with C(T), as defined in the section on calculation procedures, and replacing Tb with T in [24] .

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