How Did I Know What the Exponential Growth Curve Looked Like?

In class I showed you the shape of the exponential growth curve.  Now I would like to explain to you how I knew to draw that particular shape for that graph.

The exponential growth curve is produced anytime

dN/dt = rN and r is constant.

Remember, that making r a constant makes this equation the simplest that it can be.

If you want to know how to draw the graph then we can simply plug some numbers in to the equation and plot the results.

We will have to make up some value for r.  The simplest value  is to assume

r = 1 individual/year/individual.

All I need to do now is to calculate dN/dt for different values of N.  The simplest values of N that I can imagine are 1, 2, 3, 4, etc.  Let's start by assuming that in Year 1 N = 1 and plugging that into dN/dt = rN.

dN/dt = (1 individual/year/individual)(1 individual) = 1 individual/year

We can now add this value to the table below.


Year        N  (individual)                            dN/dt (individuals/year)
1              1                                                 1
2
3
4
5

If we started with a population size of one individual and the population increased in size by one individual during the first year then at the start of the second year the population should contain 2 individuals.  We can add that value to the table.


Year        N  (individual)                            dN/dt (individuals/year)
1              1                                                 1
2              2
3
4
5

Following the same logic that we used above you should be able to calculate N and dN/dt for both years 1 through 6.  (make sure you can do this yourself before looking at my calculations).


Year        N  (individual)                            dN/dt (individuals/year)
1              1                                                 1
2              2                                                 2
3              4                                                 4
4              8                                                 8
5             16                                                16
6             32                                                32

You should now be able to plot the following graphs using the information held in this table.

1.  How does the population size vary over time?

2.  How does the population growth rate vary over time?

3.  How does the population growth rate depend on the population size?