MATH 481, Project 5: Predator–prey models

Due Wednesday November 20

The Lotka and Volterra exchange

The Italian marine biologist Umberto D'Ancona made the curious observation that according to the available data, the percentage of selachians in the fish caught in the Adriatic sea rose sharply during World War I. He brought this to the attention of the Italian mathematician, Vito Volterra, who then produced a mathematical model to explain the phenomenon. He published a summary of his analysis October 16, 1926 issue of Nature, and a full account 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.”

Reference: An extract from Martin Braun's book

Things to do

• Explain how 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*} comes about. Hint: The idea of per capita rate of growth lies at the bottom of that model's derivation.

  Remark: Our model generalizes Lotka's and Volterra's equations, as there are no $b_{11}x $ and $b_{22}y$ terms in their model. Explain the idea behind adding those terms here. Their original model (that is, the one without $b_{11}x$ and $b_{22}y$) is known as the Lotka–Volterra predator–prey model.

  Explain how would one go about sketching the system's phase portrait. You will find that there are two qualitatively distinct cases corresponding to $\frac{a_1}{b_{11}} > \frac{a_2}{b_{21}}$ and $\frac{a_1}{b_{11}} < \frac{a_2}{b_{21}}$.

  Produce phase portraits in Maple (or the computer program your choice) to illustrate the two cases noted above.

  Draw the graph of the percentage of selachians over time to illustrate D'Ancona's data. (The data is available in Braun's textbook.)

  Explain, as Volterra did, the reason for the spike in D'Ancona's data during the World War I.

3. Sharks vs scallops

March 30, 2007 articles in the New York Times and CTV News report on the findings of a research article (download PDF) 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.

Reference

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

@article{science:sharks,
  author = {
  	Myers, Ransom A.
	and
	Baum, Julia K.
	and
	Shepherd, Travis D.
	and
	Powers, Sean P.
	and
	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. That information is needed for sorting the bibliography in the alphabetical order of the last names. The names themselves may appear in various ways depending on the bibliography style. For instance, “Myers, Ransom A.” given above may appear as “Ransom A. Myers”, or “R. A. Myers”, or in some other form.