On Reproductive Value and Fitness

As Yodzis (1981) points out, Caswell's (1980) analysis is incorrect on two counts. First Caswell misinterprets the sense in which Williams (1966), Taylor et al. (1974), and I (Schaffer 1974, 1979) claimed that maximizing reproductive value and fitness are equivalent. Another way of stating our result is as follows: let E = (E,,l,c, . . .,,) be the vector of reproductive expenditures, each defined on [0,1], that maximize (at least locally) Xh, which is the finite rate of increase of a population at stable age distribution. Further, let v = (v,, v, . . ., v,,) be the vector of reproductive values associated with X,. Consider variations in Ex and hold all other values of Ey $ . constant. Then for E, E = E,. ±+ . , 6.r small, both X, and v, will be smaller than the corresponding values which obtain if Ex = E,. Moreover, when E = t, the reproductive value, vj. , of each age class is maximized with respect to the reproductive effort, E,, at that age class. I believe that a careful reading of the various proofs that I have offered will support the claim that I have never attempted to prove other than the assertion that v.r, or more usefully B,. + P,. v,,, are maximized for each E. , maximizing X,. (Here B, is the effective fecundity of an x-year-old and P,. the probability of survival from age x to x + 1.) In particular, I have not maintained that the optimal expenditure at age x maximizes the reproductive value at other age classes, y x. It is this latter possibility that Caswell considers and quite properly rejects. Unfortunately, as Yodzis (1981) points out (see also, Charlesworth 1980:265-266), this is not the result that Williams, Taylor, and I have proposed. The second difficulty with Caswell's mathematics is that they assume that increasing reproductive expenditure at age x affects not only fecundity and survival at ages y > x, but also the values of these parameters at earlier ages, x 1, x 2, etc. Except in the case of extended parental care (for which case I have elsewhere [Schaffer 1979] given a solution), i.e., when previous years' young are still in the nest come the next breeding season, this assumption is unwarranted. This is because reproductive effort at age x is defined as the percentage of resources available at that age which are allocated to reproduction. Unless allocations to current reproduction adversely affect the survival of previous years' progeny which are still being cared for, there is no way that I can imagine changes in this years' budget affectinq the consequences of previous years' allocations. To assert otherwise would be to suppose that an individual has some fixed initial store of capital which can be spent at various ages and that Ex is the proportion of the total expended at age x. Doubtless one could construct a theory in which reproductive effort was so defined, but neither the authors cited above nor others (e.g., Gadgil and Bossert 1970, Fagen 1972) who have utilized the concept of reproductive effort as a basis for theory, have used it in this manner. To see that Caswell's (1980) mathematics make the assumption I claim they do, consider his Eq. 17 for maximizing X,. Caswell writes