A more complicated problem: Kudzu World

 

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.

1. In the absence of the other species, both daisies and kudzu grow according to the logistic equation (Eq. (4). If there were no competition, then both daisies and kudzu would have the same carrying capacity of 100 plants per acre (do you think this is reasonable?). However, Kudzu is voracious and has a growth rate 3 times that of daisies.
2. Kudzu doesn't care if there are any daisies around. I.e. its growth rate is just governed by the logistic equation and is unaffected by any pip-squeak daisies.
3. Daisies care a lot if there is any kudzu around, I.e. it's carrying capacity is directly affected by kudzu such that every kudzu plant means one less daisy plant.

Model formulation

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.

From this sketch alone, we might guess that D=0,K=0 is an unstable node, D=0,K=100 is a stable node and D=100,K=0 is a saddle node. Physically, this makes sense because it means that

• The origin is unstable because somebody always wants to grow
• Daisies can only be stable if there are exactly no Kudzu plants
• In this model, nearly all initial conditions will end up with maximum Kudzu and no Daisies.

We can now go into STELLA and make a quantitative phase portrait of the model to test our understanding. Figure 7 shows that we were right (of course). In particular, this portrait suggests that the entire region of the phase plane for K>0 and K<100 is a basin of attraction for Kudzu, i.e. any initial condition that starts in this region leads to total Kudzu domination. The problem also poses a few questions to ponder.

1. Do you think that changing the relative growth rates of Daisies and Kudzu affect their stability? What do the growth rates control?
2. Consider what happens for a trajectory that starts with lots of Daisies and only one Kudzu plant. At what point should you worry about Kudzu taking over?
3. What happens if you start with a few Daisies and more Kudzu than the carrying capacity? Whoops! what is wrong with this model?
4. In general, do you think this is a reasonable model for Kudzu invasion? How would you test it?

  
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.