# Noob: ratio between constant and variable?

jimmie
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, let's 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

## Answers and Replies

Crosson
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:

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

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

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

jimmie
Oh. Ok.

thanks

jimmie

Dr Avalanchez
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.