Let's complicate things in one way, and simplify in another. We're back with one species, and we make the following assumptions:

• there are two sexes, but only the females are of interest, as in them rests the procreative potential of the species (many mothers = many children; the same is not true for fathers);

• let n_x(t) be the number of females of age x in year t, so x and t are integers, and we can always arrange things so that they're both positive (x of course is positive anyway);

• the females are assumed to start breeding from age x = 1, and to continue breeding each year of subsequent life up to and including age x = w (so from age x = w+1 onward they drop out of this mathematical model and we'll just assume they retire to the Ozarks);

• let P_x = probability that a female of age x survives to age x+1. P_x is the fraction of x year olds likely to survive, so that means

  n_x+1(t+1) = P_x n_x(t).

  That is, if we observe n_x(t) x year olds in some year t, then in the next year (t+1) only a certain fraction will still be around, not x+1 year olds.

• Each of the n_x(t) females of age x is assumed to annually give birth to a certain number of females, but they're of little interest unless they survive to be 1 year olds, at which point they start breeding. Let f_x = average number of females born of x year olds that survive to age 1 (this may be a fraction). Therefore, if we start with n_x(t) females of age x in year t, then in the next year (t+1) we assume there will be about f_x n_x(t) of their daughters of age 1 still around. So the total number of 1 years olds in year t+1 would be

  n₁(t+1) = f₁ n₁(t) + f₂ n₂(t) + ...
  + f_w n_w(t).

  This is our final recursion relation (meaning it relates a new value of a variable to some old values). Note that n_x+1(t+1) = P_x n_x(t) starts with x = 1, not x = 0, so we have w recusion relations in all.