# Two paradoxes of reproductive fitness

A simple conception of reproductive fitness is that the reproductive fitness of an organism is measured by the number of offspring that it has that survive and reproduce in turn. This conception is clearly inadequate; an organism may have many offspring whose offspring in turn fail to produce reproducing offspring in turn. What we want to look at is the number of descendents that an organism has over time. This is the relevant factor because in the long run reproductive fitness that ultimately selected for.

For sexually reproducing organisms we must take into account that an individual organism only makes half of the genetic contribution to each of its offspring. Iterating, we conclude that in turn it contributes one quarter of its genes to each of its offspring's offspring and so on so that in the n'th generation of descendents the contribution to individual descendents is 1/2**n. Accordingly the average number of copies of an organism's genes in the n'th generation is

(1) RF(O,n) = D(n)/2**n
where D(n) is the number of descendents that the organism has in the n'th generation and RF(O,n) is the reproductive fitness of organism O in the n'th generation. We now come to paradox 1.

Paradox I - Over time reproductive fitness decays to zero.

The world being finite, populations are bounded in the long run which is to say that D(n) is bounded. However the term, 1/2**n, becomes arbitrarily small with increasing n. Hence, in the long run RF(0,n) goes to zero.

The difficulty here is or should be obvious. In the long run copies of the organisms genes will be scattered throughout the population so that an organism in the n'th generation can get some genetic contribution from each of its parents. The fault in equation (1) is that it assumes unlimited outbreeding. Clearly we need to consider the mean number of copies of the organism's genes in generation n, taking into account the contribution of both parents. For any particular descendent, its inheritance from the original ancestor is (on average) (m+f)/2 where m is the maternal and f the paternal inheritance ratios. This brings us to paradox 2.

Paradox II - Incest optimizes reproductive fitness

The argument is quite simple. Siblings (from a non-incestuous union) have 50% of their genes in common. If they mate their offspring have 75% of their genes in common with each parent and with other. If two unrelated individuals mate and have two offspring each parent has on average one copy of their genetic material in the next generation whereas if two siblings mate they each have 1.5 copies of their genetic material present in the next generation. Clearly incest optimizes reproductive fitness.

There are certain fallacies in these paradoxes. The reader may enjoy finding them.