Generalizing the Equations
| n'(0) = p * n(0)/2 | (1.) |
| n(1) = n(0)*[1 + p/2] | (2.) |
The equations above are written in terms of 2 specific years, the year 0 and the year 1. But the equations themselves contain nothing which restricts their applicability to those two years. The equations apply in general to any 2 consecutive years. The quantity on the left-hand side of the equation describes the number of rabbits in the year after the year referred to by the terms onthe right-hand side of the equation. So, if we call the year referred to by the right-hand side of the equation the year t , then the year on the left-hand side of the equation is t+1.
Thus, in terms of a general year t, our 2 equations become
| n'(t) = p * n(t)/2 | (3.) |
| n(t+1) = n(t)*[1 + p/2] | (4.) |
| n'(t) = 2 * n(t)/2 = n(t) | (5.) |
We can now use this specific version of the rate equation to get a form of the iterating equation specific to our choice of p = 2:
| n(t+1) = n(t) + n(t) = 2 * n(t) | (6.) |
To use this specific iterating equation we start with the number of rabbits in the year 0, which we chose to be 4. So in the year 1
n(1) = 2 * 4 = 8
Now we can simply repeat the use of Eq. 6 to calculate the number of rabbits in the year 2.n(2) = 2 * 8 = 16
At this point the calculation get repetitive, so instead of doing is by hand we can program a computerto do it.
To see how a computer can iterate this rate equation try the "Rabbits Applet" below.
You should try changing the parameters of the model to see how the data are affected. You can change the number of rabbits you start out with in year 0, or you can change the number of babies per litter, the procreation coefficient.
Once you have understood how this simple model works, go on to the more complicated model