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MathBio: Gearing up for Test 2

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:

Horseshow crabs 1a   Horseshoe crabs 1b

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.

\dot S = - \alpha SI + \gamma R

\dot I = \alpha SI - \beta I

\dot R = \beta I - \gamma R

Set \beta = 2, \gamma = 1 and impose the constraint that S + I + R = 1.  Rewrite the system as a two dimensional system (in S and I only) and determine a condition for \alpha such that there exists an endemic equilibrium (meaning an equilibrium (S_*,I_*) where S_* > 0 and I_* > 0.  Classify the type of the endemic equilibrium.

SOLUTION

There is an endemic equilibrium if \alpha > 2.  CHECK me on this next part!!!  I’m getting that the endemic equilibrium is a stable spiral if \alpha \in (2.0294, 35.9706)

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 a.

\dot x = x ( 2 -x - y)

\dot y = y (a - 2x - y)

  • Sketch the x and y nullclines in three cases:  a = 1, 3, 5.  In each case, note whether there is an equilibrium in the first quadrant (a co-existence equilibrium).
  • For what values of a does this system have a coexistence equilibrium?
  • When the coexistence equilibrium exists, what is its type?

SOLUTION

The x- and y- nullclines for this problem are pictured below.

Nullclines of the Challenge problem

Nullclines of the Challenge problem

There is only a coexistence equilibrium if a \in (2,4). This equilibrium has coordinates (x_*,y_*) = (a-2, 4-a).  After computing the Jacobian and evaluating at the fixed point, we see that the trace of the Jacobian is -2 for all values of a, while the determinant is -(a-2)(4-a) which is always negative.  We conclude that, when it exists, the coexistence equilibrium is always a saddle node.

 

MathBio: Midterm 2, Wednesday, April 2

Our second midterm will be Wednesday April 2.  The test will cover topics from Chapters 3 and 4.

More details will be posted soon, but the main theme is:

Qualitative Analysis of two-dimensional systems in both discrete and continuous time.

A non-exhaustive list of topics includes:

  • Biological Modeling concepts
    • Age-structured models (e.g., Juvenile-Adult)
    • Epidemic models (SIS, SIR, SIRS)
    • Multi-species models (Predator-Prey, Niche Competition, Host-Parasitoid)
  • Linear Algebra concepts
    • Writing dynamical systems as Matrix-Vector equations
    • Eigenvalues and Eigenvectors
  • Analysis of linear 2d discrete-time systems
    • The Jury Conditions for Stability
    • Computing Stable Age Structures
  • Analysis of nonlinear 2d discrete time systems
    • Finding equilibria
    • Computing the Jacobian and evaluating at the equilibria to determine stability
  • Analysis of linear continuous-time systems
    • The Trace-Determinant Plane
    • The three forms of general solutions when the eigenvales are 1) real and distinct; 2) real and repeated; 3) complex.
  • Analysis of nonlinear continuous-time systems
    • Finding equilibria
    • Using nullclines to sketch flow in the phase portrait
    • Computing the Jacobian and evaluating at the equilibria to classify type

Reminder: No office hours today, Tuesday 3/18