## 3. What the model looks like

A plot of the logistic growth curve looks like this:

Notice that the population starts with one individual, and eventually reaches an equilibrium value, in this case about 2000 individuals (the value of K set for this example). The rate at which the slope of the curve **increases** initially is mirrored by the rate at which the slope **decreases** as the population approaches the carrying capacity. The greatest value of the slope of the curve occurs just after ten days in this example, when the population is exactly half of the carrying capacity (0.5K).

In this example, the value of r has been set so that initially the population doubles once per day. If r had been larger, the population would have increased more rapidly initially, and would have reached a level equal to 0.5K earlier. It would then have seen a faster decline in growth rate close to carrying capacity, which would also be earlier than in this example. The curve would look just like this illustration, but compressed horizontally. Conversely, a low value of r would make things happen more slowly, if K were left unchanged. The curve would appear to be stretched out horizontally.

We can explore the effects of changing r*by looking at the time taken to reach a population of 0.5*K:

K (number - constant) | 2000 | 2000 | 2000 | 2000 |

R (see section 2) | 1.5 | 2 | 3 | 4 |

r (d^{-1}) |
0.4055 | 0.6931 | 1.0986 | 1.3863 |

Time to achieve 0.5K(d) |
c. 18.5 | c. 10.5 | c. 7 | c. 5.5 |

The shaded column uses the same parameter values as the example illustrated in this section.

Similarly, we can examine the effects of changing K whilst keeping r constant. Altering carrying capacity affects the speed at which the population stabilizes – for a given value of r, low K means that the population stabilizes early whilst high K means that it takes a longer time to equilibrate. However, the effects of quite large changes in K are less marked, especially when r is high as in the illustrated example:

K (number) | 500 | 1000 | 2000 | 5000 |

R (see section 2) | 2 | 2 | 2 | 2 |

r (d^{-1}) |
0.6931 | 0.6931 | 0.6931 | 0.6931 |

Time to achieve 0.5K(d) |
c. 9 | c. 10 | c. 10.5 | c. 12.5 |