If you a looking for problems to work on to prepare for Midterm 2, this post will be the place to look. This will be updated several times over the next few days.
Challenge problems for discrete-time systems.
Solutions to the important parts of the Horseshoe Crab project are written up here:
For more practice, I recommend 3.3 #5 from the book.
Challenge problems for continuous-time systems.
The first and most important place to find problems for continuous time systems is in my supplemental lecture notes: Systems of ODEs (pdf). See in particular problems 4, 5, 6 and 7. Solutions are included already for 4-6. I will post a solution to 7 on Monday.
Challenge problem on epidemics.
Set , and impose the constraint that . Rewrite the system as a two dimensional system (in and only) and determine a condition for such that there exists an endemic equilibrium (meaning an equilibrium where and . Classify the type of the endemic equilibrium.
There is an endemic equilibrium if . CHECK me on this next part!!! I’m getting that the endemic equilibrium is a stable spiral if
Challenge Problem for Nullcline Analysis
Consider the following system of ODEs which are from the class of Niche Competition models introduced in class. (Both species exhibit logistic growth models when alone and have negative mutual interactions.) In this case, we have one unknown parameter .
- Sketch the and nullclines in three cases: . In each case, note whether there is an equilibrium in the first quadrant (a co-existence equilibrium).
- For what values of does this system have a coexistence equilibrium?
- When the coexistence equilibrium exists, what is its type?
The x- and y- nullclines for this problem are pictured below.
There is only a coexistence equilibrium if . This equilibrium has coordinates . After computing the Jacobian and evaluating at the fixed point, we see that the trace of the Jacobian is for all values of , while the determinant is which is always negative. We conclude that, when it exists, the coexistence equilibrium is always a saddle node.