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Instead of some parents having lots of kids, if the number of kids were spread more evenly to parents having few kids and people who have no children, would it introduce more genetic variability in our species in general?

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Well the obvious answer is of course yes of course it would. That is the essence of natural selection. Variability leads to different phenotypes and selection for and the amplification of beneficial traits. Evolution. Except if there is no selection (everyone has the same amount of kids) no traits are amplified so everything in a large population is static. Until you introduce mutations and that increases variability. Eventually each individual would have such a large degree of variation from the next that I would assume speciation would be a very imminent result of any form of seperation or selection, which is against your original premise.

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  • $\begingroup$ This is a good answer, and the only thing I would add is that the question does not mention whether all children have an equal chance to grow up and reproduce. Given that accidents and illnesses might not be entirely random and might have genetic dependencies, natural selection would still be free to operate in a society where fecundity was evenly spread. $\endgroup$ – Jonathan Moore Aug 28 at 15:29
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To answer this question we need to recourse to the concept of "effective population size" ($N_e$). This is the number of individuals which actually counts in maintaining a population and generally does not coincide with the number of living individuals (the census size $N$). $N_e$ was defined by Sewall Wright as "the number of breeding individuals in an idealized population that would show the same amount of dispersion of allele frequencies under random genetic drift or the same amount of inbreeding as the population under consideration".

In an idealized population with constant size, the chance of generating offspring is the same for all individuals, in such a way that the distribution of the number of offspring per individual is Poisson. In this situation $N_e = N$. In real populations, of course, the number of individuals of the two sexes that actually reproduce may be highly unbalanced, there is a large variation in fertility of different pairs, the number of individuals fluctuates from one generation to another, etc. All these factors lead to deviations such that $N_e < N$ (in most cases $N_e \ll N$).

The effect of variation in progeny size on $N_e$ is modeled by the equation $$N_e = \frac{4N - 2}{V + 2},$$ where $N$ is the total number of individuals (half males half females), and $V$ is the variance in progeny size per mating pair. It may be noted that if $V = 2$, which is the Poisson variance of a population with constant size, $N_e \approx N$ for reasonably large $N$, whereas $N_e$ decreases as $V$ increases.

Therefore, the answer to the question is yes, genetic variability is maintained at a higher level if the number of children is distributed more evenly among pairs.

An interesting point to note is that if the variance in progeny size is less than Poisson, we find $N_e$ > $N$, a result that may appear surprising (if $V = 0$, $N_e \approx 2N$). This suggests a strategy to maximize the effective size of captive populations of species at risk; by sampling a male and a female born from different pairs to mate for sustaining the population, the genetic variability transmitted to the offspring is maximal and the inbreeding is minimized.

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