Project 6, MATH 481, Spring 2008
Population dynamics: Sharks, rays and scallops

Assignment of 2008–04–14

Here are suggestions about what to write about. Feel free to deviate from these if you can think of other interesting things to say.

• Introduce a model of three interacting species of sharks, rays and scallops, as discussed in class and the handouts. Sharks prey on rays. Rays prey on scallops.
• Introduce harvesting, the rate of harvesting of each species being proportional to that species' population.
• Find the system's equilibria. Specifically focus on the nontrivial equilibrium where all three components are nonzero.
• Investigate the effect of harvesting on the equilibrium. In particular, note the effect of harvesting of sharks on the scallop's equilibrium population. Correlate this to the findings of Myers et. al.
• Make up numbers for the coefficients of the system. Verify that the three components of the nontrivial equilibrium are all positive. Make up numbers for the initial data. Plot the solution curves x(t), y(t) and z(t) against time t. Decide if you want to show the three curves superimposed on one diagram (in which case it would be a good idea to use three different colors for the curves) or if you want to show them as three separate graphs.
• Additionally, produce a graph of the corresponding orbit in the (three dimensional) phase space.
• Compute the system's Jacobian matrix and evaluate it at the nontrivial equilibrium using your previous choice of numbers for the coefficients. Compute the eigenvalues of the linearized system and verify that they agree with the observed behavior of the orbit in the phase space near the equilibrium.

Reference

For your convenience, here is the amsrefs entry for the cited article. In your article you may also cite the NYT report, if you wish.

  \bib{meyers:sharks}{article}{
    author = {Myers, Ransom A.},
    author = {Baum, Julia K.},
    author = {Shepherd, Travis D.},
    author = {Powers, Sean P.},
    author = {Peterson, Charles H.},
    title  = {Cascading Effects of the Loss of
              Apex Predatory Sharks from a Coastal Ocean},
    journal= {Science},
    volume = {315},
    pages  = {1846--1850},
    date   = {2007-03-30},
  }

Due date

This assignment is due on Wednesday April 23.