Spring 2017, MATH 481, Project 5
Predator-Prey Models and Trophic Cascades

Due Friday April 28

Outline

• Explain the derivation of the predator-prey model \begin{align*} \frac{dx}{dt} &= ( a_1 - b_{11} x - b_{12} y)x, \\ \frac{dy}{dt} &= (-a_2 + b_{21} x - b_{22} y)y. \end{align*}
• Describe the essential features of the system's phase portrait and its equilibria. Qualitatively, there are two distinct cases, depending on whether $a_1/b_{11} < a_2/b_{21}$ or $a_1/b_{11} > a_2/b_{21}$. Show a sample phase portrait for each case.
• Introduce harvesting proportional to the population of each species. If the system has a nontrivial equilibrium (i.e., an equilibrium where both populations are positive), how does harvesting affect it? This relates to Volterra's explanation of the increase of the fraction of selachians in the fish caught in the Adriatic Sea during the WWI.
• March 30, 2007 articles in the New York Times and CTV News report on the findings of a recently published research article in Science which points to a link between the overfishing of large sharks in the northern Atlantic, and the destruction of the bay scallop fisheries off the North Carolina coast.

  Set up a mathematical model of a 3-species trophic cascade and analyze it to explain the article's findings.

The Lotka and Volterra exchange

Vito Volterra published a summary of his analysis of the surge in the fraction of the Adriatic selachian population in the October 16, 1926 issue of Nature. The full account was published in a 1927 report titled Variazioni e Fluttuazioni del numero d'individui in specie animali conviventi.

In a Letter to the Editor in the January 1, 1927, Alfred Lotka pointed out that the October article duplicates parts of the analysis in his book that was published in Baltimore in 1925.

Volterra responded in the same issue, acknowledging Lotka's remark, and wrote: “In this I recognize his priority, and am sorry not to have known his work, and therefore not to have been able to mention it.”

Their predator-prey model is now known as the Lotka–Volterra model, although most commonly the model is presented without the overcrowding effects, that is, with $b_{11} = b_{22} = 0$.

Supplementary literature

You may be interested in the following articles which one way or another are related to trophic cascades:

• The 1998 article from Science reports the adverse effects of the increased killer whale population on the sea otters of Western Alaska, and the consequent changes in the sea urchin and kelp biomass.
• The April 26, 2010 New York Times article and video is about Glover’s Reef, a protected marine reserve off the coast of Belize. Fishing is strictly prohibited in this 17,500-acre zone in the Atlantic.
• A major civil engineering project in the late 1800s reversed the flow of the Chicago River away from Lake Michigan, and through a series of canals and locks, connected it to the Mississippi River.

  Asian carp, a voracious non-native fish which has established itself in the Mississippi, has been migrating upward through the canals and now threatens to invade Lake Michigan and the other Great Lakes.

  The January 2, 2010 New York Times article reports on the controversy surrounding the proposal to close the canals to prevent the invasion.

Reference

Here is an amsrefs entry for the 2007 Science article. You may want to add other citations as needed.

\bib{science: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},
    date = {2007-03-30},
   pages = {1846--1850},
}

Important! How to enter an author's name

In the bibliographic entry shown above, note that each author field consists of the last name, then a comma, then the rest of the name. You should always supply the names in that order.

That tells LaTeX which is the last name and which is the first. What it does with that information is something else. For instance, the name “Myers, Ransom A.” given above may appear as “Ransom A. Myers”, or “R. A. Myers”, or in some other form. LaTeX and its documentclass will decide how. Your job is to supply the raw data in the proper form: last name, comma, the rest.

Author: Rouben Rostamian