## When Will Evolution Just Maximize the Malthusian Parameter Average Offspring Number

The results from the previous subsection can be used to characterize the ecological scenarios for which evolution will just maximize the Malthusian parameter or lifetime offspring number. Here 'just maximizing a function of X and E' should be interpreted as maximizing that function by varying X for an unspecified choice of E (the latter as reflection of the absence of any mention of E in the usual statements in the nonepidemiological literature). Under the presupposition that the community dynamics engenders constant environments so that the Malthusian parameter r and the lifetime offspring number R0 are well defined, it can be proved that evolution just maximizes r if and only if

(D) the combination of life histories and ecological embedding is such that r can be written as r(X|E) = g (r(X|E0), E) for some function g that increases in its first argument, and E0 some fixed, but otherwise arbitrary, environment, and evolution just maximizes R0 if and only if

(E) the combination of life histories and ecological embedding is such that ln(R0) can be written as ln (Ro(X|E))= g(ln(R0(X|Eo)), Eo) for some function g that increases in its first argument, and E0 some fixed, but otherwise arbitrary, environment.

In contrast to the criterion for the existence of an optimization principle, the present criterion is relatively easy to check in specific situations.

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