# Noob: ratio between constant and variable?

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

## Answers and Replies

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
Science Advisor
The actual ratio is of the order 10-9 (well 10-8) so the ratio will be unchanged for quite a long period of time. (In the social sciences if you can get to + or - 5 percent, you are generally doing well. In epidemiological studies, I believe that they shoot for better accuracy.) Many ratios used in demographics and economics, labor force participation ratios, unemployment rates, proportion of US household with an income of \$100,000, rates of household mobility, proportion of owned housing units and so forth can tend to vary with time. And, sometimes, it's the numerator that changes more than the denominator.Also, some of these ratios remain quite constant, like mobility rates (Mobility refers to moving from one dwelling unit to another). Indeed, to do things right, you need to include what's coming in -- births, migration -- and going out-- deaths and migration. In computer simulations, generally you use difference equations relating a state to it's plus and minus sources, which leads, in simplest form, to exponential variations. As the good Dr. points out, there are cases in which the plus and minus rates are population dependent. (For an exhaustive account of such models see the classic by M.S. Bartlett, An Introduction to Stochastic Processes, written in 1962, published by Cambridge University. It is still a very valuable text, in spite of its age.)

Regards,
Reilly Atkinson