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