When will Long Term Evolution Maximize Some Fitness Related Measure

Given the stress on fitness maximization in the literature, it is relevant to know when there do exist properties oftypes that are maximized at CSSs. This is the case if and only if'the trait values affect fitness effectively in a one-dimensional monotone manner'. The term 'effectively' here means that the specified properties only need to pertain to the range of fitness values closely surrounding the change from negative to positive. More precisely, the following results hold, with X denoting the potential resident vectors, E the realizable environments, and the R the real numbers:

An ecoevolutionary model is governed by an optimization principle if and only if

(A) there exists a function ^ : X ! R and a function g : R x E ! R , increasing in its first argument, such that sign p(X|E) = sign g(^(X), E) Condition (A) can be proved equivalent to:

(B) there exists a function ^ : E ! R and a function h : X x R ! R, decreasing in its second argument, such that sign p(X|E)= sign h(X,^(E))

which can be paraphrased as 'the environment acts effectively in a one-dimensional monotone manner'.

Relations [11] and [12] can be related to each other by the observation that if an optimization principle exists,

(C) it is possible to choose the functions 0 and such that sign p(X|E) — sign h(0(X) - 0(E)) [13]

where 0 and -0 are connected through the relation

Of course, results (A)-(C) hinge on the interpretation ofthe term 'optimization principle'. The latter should be interpreted as a function from trait values to real numbers such that for any possible constraint on the traits the ESS(s) can be calculated by maximizing this function. The proviso in the previous sentence mirrors the usual practice of combining an optimization principle, derived from the population dynamics, with a discussion of the dependence of the evolutionary outcome on the possible constraints. What matters here is that, while condition (A) is close to trivial, the equivalent condition (B) and relation [14] provide a useful tool for either deriving optimization principles or proving the nonexistence of such principles for large families of ecoevolutionary models.

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