Why the assumption that things are all exponential?
Put another way, there is no number of children per couple C such that a population initially composed of N individuals will always add X>1 number of children per year in perpetuity.
If every person has one kid (or alternately, every couple two), then the growth rate is zero and the population constant. Constant population is exponential (exponent = 0).
If the number per couple is below two, then population declines exponentially. Fewer and fewer people die each year but also fewer are added. Eventually the population hits zero
If the number is more than two, then more and more are added each year