Exponential Decay

Exponential Decay

Suppose that at the end of every year (at 11:59:59.999 pm on 31 Dec), a population loses a fraction $\mu$ of its members.

As an example suppose that $\mu = 0.1$, so that 1/10th of the surviving population disappears at the end of each year.

Over 5 years an initial population of 100,000 would change like this if $\mu = 0.1$:

Year Population at Start Losses at End
0 100 000 10 000
1 90 000 9 000
2 81 000 8 100
3 72 900 7 290
4 65 610 6 561
5 59 049

We could make the change smoother if we reduced the population by ${\mu}/2$ every half year. In our example ${\mu}/2 = .05$ and we would have

Year Population at Start Losses at End
0 100 000 5 000
0.5 95 000 4 750
1 90 250 4 512.5
1.5 85 737.5 4 286.9
2 81 450.6 4 072.5
2.5 77 378.1 3 868.9
3 73 509.2 3 675.5
3.5 69 833.7 3 491.7
4 66 342 3 317.1
4.5 63 024.9 3 151.2
5 59 873.7

Smoother still would be if we reduced the population by ${\mu}/4$ every quarter year. In our example ${\mu}/4 = .025$ and we would have

Year Population at Start Losses at End
0 100000 2500
0.25 97500 2437.5
0.5 95062.5 2376.6
0.75 92685.9 2317.1
1 90368.8 2259.2
4.75 61814.1 1545.4
5 60268.8

For example,

(1)
\begin{align} s(x) = e^{-{\mu}x} \end{align}

[[experimental notes]]

page revision: 25, last edited: 24 Aug 2010 15:19