## 2. Real-world growth models - why geometric series don't work

If we have a large population of bacteria which are dividing at an average rate of once per day, we would expect the population to double each day as described in the introduction. This time, we will start with 100 bacteria, and the population change can be represented in a table like this:

Time (days) | Starting population | New bacteria produced |

0 | 100 | 0 |

1 | 100 | 100 |

2 | 200 | 200 |

3 | 400 | 400 |

4 | 800 | 800 |

However, in a large population individual bacteria can divide at any time. What happens if we use the geometric series to calculate the population size every 12 hours (half of the doubling time)? We would expect that half of the bacteria would have divided in that interval:

Time (days) | Starting population | New bacteria produced |

0 | 100 | 0 |

0.5 | 100 | 50 |

1 | 150 | 75 |

1.5 | 225 | 113 |

2 | 338 | 169 |

2.5 | 508 | 254 |

3 | 762 | 381 |

3.5 | 1143 | 572 |

4 | 1715 | 857 |

In this second calculation, the growth rate is still the same (once per day) but we have evaluated the series at 12 h (0.5 day) intervals and the population appears to grow faster. This is because the bacteria added at each interval contribute to the 'current' population that will divide in the next interval.

So which model is correct? The answer, sadly, is neither! The bacteria are the same, each one dividing once per day. But the population calculations are dependent on the time interval used. If we calculate the geometric series using different time intervals, we see that we approach a stable value for population growth as the interval gets smaller and smaller. This can be seen if we use the population after one day as a benchmark:

Time interval for calculation (days) | Number of bacteria after one day |

1 | 200 |

0.5 | 225 |

0.25 | 254 |

0.1 | 260 |

0.05 | 266 |

0.02 | 269 |

0.01 | 271 |

0.005 | 271 |

Decreasing the time interval in a geometric series like this does two things – it adds new bacteria to the population more quickly (increasing the current population) but also decreases the population increase at each time-step (because the time interval decreases). So we expect that as we shorten the time interval more and more, the population growth curve will gradually settle down to the same value.

It is easy to evaluate the geometric series in a spreadsheet, but it becomes tedious to calculate populations over long periods of time with very short time intervals. More importantly, the growth rate over very short intervals is not a simple fraction of the average growth rate. Instead of the simple geometric series, we need a population growth model that can be evaluated for any time and is independent of time interval.