Thanks for the links. However, it seems that the connection isn't that strong, e.g. from the last article:
"we should not be confident that reducing mortality will always reduce population growth . Methods other than reducing mortality seem more effective in reducing population growth."
"the most effective means to achieve decreases in fertility and population growth in Sub-Saharan Africa would be to invest in family planning services and female education."
But except for Sub-Saharan Africa it seems to be working well so I feel kinda convinced :)
thanks to you both, that looks pretty convincing