Okay, enough words, time to demonstrate these ideas with one more worked out problem. Let's put together a simple minded model for invasion of the killer Kudzu. In our model we will have two species, daisies and Kudzu (or crazy ants and everybody else) and they will obey the following rules.
Let D(t) be the number of daisies per acre. K(t) be the number of Kudzu plants and r be the growth rate of daisies. The only big trick to this problem is that the carrying capacity of daisies depends on the number of kudzu plants and is simply 100-K. With these definitions we can write down our Kudzu world model as
Okay, let's analyze this model starting with Kudzu because it is exactly the same as our logistic growth in section 2.2. A useful first step in these problems is to first plot all the lines along which growth in each direction is zero (these curves are called the nullclines. For Kudzu, there is no growth if K=0 and K=100 as before. Next we'll look at the nullclines for the Daisies. You can check for yourself that there is no daisy growth if D=0 (no daisies) and D=100-K, i.e. if the daisies are at their carrying capacity that is determined by the amount of Kudzu. Confirm for yourself (or use a graphing calculator) that D=100-K is a straight line (see Figure 6).
Next we find the fixed points where both change functions are zero at the same time. But these are just where the nullclines cross! So you can check for yourself that there are only three fixed points in the problem at (D=0,K=0), (D=0,K=100) and (D=100,K=0). Question: what is the physical meaning of the three fixed points?.
The final step before STELLA-izing is to sketch a few change arrows on the diagram to guess at the stability of the different fixed points. We simply go to a few choice locations on the phase plane and ask ``at this point, are Daisies increasing or decreasing and is Kudzu increasing or decreasing and plot a small arrow showing how we expect the system to move''. Figure 6 shows the results of all the steps so far (and we will reconstruct this in class).
Figure: A quick sketch of Kudzu world, showing 4 nullclines, 3
fixed points and some change arrows. Red nullclines are lines
where there is no daisy growth . The green
nullclines are where kudzu growth is zero ( ).
The fixed points occur where both growth rates are zero (which
is where the red and green nullclines intersect.
Figure 7: Quantitative phase portrait for the Kudzu invasion
problem for a value of r=1 a carrying capacity of 100, and a
set of trajectories starting with just one kudzu plant and a
wide range of Daisy populations. Note that all initial
conditions for K<100 end up with nothing but kudzu.