# Noob: ratio between constant and variable?

## Main Question or Discussion Point

OK, so I'm not a math major. :( or is it :) (I guess that's philosophy)

Anyway, here is what i'm stumped on.

Population. It is always changing.

So, lets say that only 1 person in the population has a particular skill that no other person has.

What would the correct ratio be of persons having that particular skill?

Can't be 1:6,400,000,000 or whatever the current population is, because by the time you finish reading this statement, the population has changed.

jimmie

Because the population changes continuously, you can represent the ratio you desire as a continuous function of time. If b is the birth rate, P is the current population, and e is the constant 2.71828..., then the ratio could be modeled the following way:

$$Ratio at time t = P e^{-bt}$$

Is it possible to express the ratio in common terms as with other ratios?
1:6,500,000,000 type of thing?

jimmie, the ratio has to be expressed as a function of time in order to be as accurate as can be.

Oh. Ok.

thanks

jimmie

A population model is actually a differential equation (like Crosson made an attempt).

A first order approach would be a linear differential equation, the malthusian population model (posited in 1798 by the British economist Thomas Malthus): P'(t)=rP(t), with P(0)=6,400,000,000; where P(t) is the number of people at a given moment, r is a growfactor (r>0 means growth, r=0 means constant, r<0 means decrease).

Your ratio would be (assuming no one else with that special skill comes onto play) 1/P(t).

Of course, the exponential growth is only acceptable over a short periods of time. Competition for food, local factors,... will tamper the growth of a population. A second order approach is a non-linear approach, the population model of Verhulst (1837):
P'(t) = eP(t) - sP(t)², P(0)=6,400,000,000; with s<<e.

reilly