Thanks for an interesting article; a few thoughts.

First, least important, the term "fertility rate" used throughout literature is a bit unfortunate; what is meant quite often is "reproduction rate". To me, fertility is about the ability for humans to have children and is influenced by many factors, e.g. biological such endocrine disrupting chemicals and toxins. Fertility will influence the reproduction rate.

Second, there are strong correlations between various quantities: reproduction rate, GDP/capita, energy/capita, CO2/capita, etc, all connected into a dynamic system without clear causality. So more affluence leads to both a decline of the reproduction rate and an increase in pollution which in turn will decrease fertility (men and women). All together leads to population decline and hence less consumption & production, less pollution, too, but does not go well with a growth based economic system: think pension funds, or any investment that links expectations for the future to the present. As mentioned on other occasions, a controlled economic contraction would be preferable (including decreasing world population); but I am afraid we seem to be geared up for growth & collapse. Restraint is not popular, growth is a shown by recent elections in various countries and executive orders that followed.

On population modelling, I found Tom Murphy's work interesting: Peak Population Projections & Watching Population Bomb.