Models with Selection on Both Viability and Fertility (Fecundity)
M. Frank Norman, 10/1/2008
This note concerns models for evolution under selection on the basis of viability (probability of survival to reproductive age) and fertility (related to number of offspring) in a large, random mating population. It is an introduction to the spreadsheet
that calculates and graphs the trajectories of these models for arbitrary parameter values, including values where population mean fitness declines.
We assume that there are only two alleles, A1 and A2, hence three genotypes, a = A1A1 , b = A1A2 , and c = A2A2 , with viabilities sa, sb, sc, fertilities ta, tb, tc, and genotype frequency distribution pa, pb, and pc in zygotes (newly conceived individuals) of some generation. This distribution is the same for males and females.
The viability selection process leads to a modified genotype frequency distribution ps, where psg = pgsg/sum(phsh), g = a, b, c. (Sums are always over repeated indices.) There is then random mating among surviving adults, and a mated couple of gentypes g×h produces, on the average, f(tg,th) offspring, whose genotypes are distributed according to the usual Mendelian rules. The expected number, ng , of offspring of genotype g can then be calculated from a table like those used in the derivation of the Hardy-Weinberg law, and these numbers can be divided by their sum to obtain the zygotic genotype distribution in the next generation.
In this model, the fitness (expected number of offspring) of an individual of genotype g is
wg = sg sum(f(tg,th)psh)
where sg takes account of survival or non-survival and the sum is an average (using post-survival probabilities) of couple fertilities over various possibilities of g’s mate. Population mean fitness is the average (using pre-survival probabilities) of these,
w = sum(wgpg).
We focus on the cases of additive and multiplicative fertility, f(t,u) = t+u and t×u. The corresponding mean fertilities can be shown to be
additive w = 2 sum(pgsgtg) = 2 sum(pgsg) sum(psgtg)
multiplicative w = sum(pgsgtg) sum(psgtg) = sum(pgsg) (sum(psgtg))2 .
Norman, M. Frank, A very simple model for declining mean fitness, Journal of Bioeconomics, 2005, 7, 157-160,
I showed that mean fitness could decline in the additive case, but claimed that this could not happen in the multiplicative case, since that case was equivalent to a pure viability selection model with viabilities sgtg , and mean fitness never declines in one-locus viability selection models. I now see that this claim was not entirely correct. The claimed equivalence is well known for trajectories of gene and genotype frequencies, but the formulas for mean fitness are not equivalent, and, in fact, the multiplicative fitness model can exhibit declining mean fitness for parameter values not very different from those that produce declining mean fitness in the additive model.
The spreadsheet calculates and graphs trajectories for arbitrary parameter values for both additive and multiplicative models, and highlights parameter values for which mean fitness declines.