Table 1: How Changes in Fertility and Mortality Affect Fitness Effect Additive Reproductive Discount Relationship
on change value factor to donor Donor
A. fert. Am) 1 ene) 1 B.mort. AP yh Ala) +-der) i Recipient
C. fert. Am) 1 e—P(2\) +7) r D.mort. AP() y) eo Ale +7-+de) r
Notes: The altruist allele will increase (decrease) in frequency if the sum of row products is positive (negative). The notation is defined in the text. For simplicity, I assume that the sex ratio at birth is unity, that effects on fertility are brief, that these effects are small enough that second-order terms in Am and AP can be ignored, and that a single recipient is affected by each altruistic act. Source: Rogers [14, Table 1]
the recipient, and v denotes the reproductive value (R. A. Fisher, 1958). It is defined by
_ ae e PY g(y)Mmg (y) 7 e~ Pl (x)
(4)
where p is the rate of population growth, [,(y) the probability of living to age y, m,(y) the ex- pected number of offspring produced at that age, and the subscript g indicates the individual’s sex. The reproductive value can be interpreted as the expected present value of an individual’s future contributions to the gene pool.
A gene that encourages the donor to undertake this action will be favored by natural selection if the sum of the row-products in table 1 is positive, or disfavored if that sum is negative.
2.2.2 The MRS in fitness
An interaction is selectively neutral—having no effect on fitness—if the sum of row-products in table 1 is zero, Le. if
+ Am eRe Ot p 4 A PA) yA POM +) 9 (5) Here, I have assumed that effects on mortality are brief so that dr ~ 0. When this equation holds, the interaction (or investment) described above moves us along a fitness isogram. Thus, the equation holds the key to the slope of this isogram, the MRS in fitness. But before proceeding, it will be useful to recast the equation in terms of changes in consumption.
I now assume that fertility and mortality are both differentiable functions of consumption. P = P(z,k)
m
m(a, K)
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