anywhichway

Natural generative processes come down to growth force. When you write them down as the results of your most basic observations of them, like "Growth is exactly proportional to the population size" you end up putting everything in terms of these "forces" which yields a differential equation p'(t) = k p(t) which has a solution of p(t) = c e^(kt).

There are a number of basic tweaks that reflect real life situations that can quickly make these functions unsolvable by elementary functions. One example are time-varying carrying capacity. If you're on an island that generates a limited amount of resources each year you get the basic carrying capicity logistic function p'(t)=p(t)(1-p(t)) which is solved by p(t) = 1/(1+c e^t). But if that your population has a delayed affect on the available resources you get p'(t)=p(t)(1-p(t-t1)) which can't be expressed as elemantary equations. http://en.wikipedia.org/wiki/Logistic_function#Time-varying_...

Also, preditor-prey models which occur naturally where you have two populations that depend on each other which can't be reduced to elementary functions either:

http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation