MATH 481, Project 5: Of Sharks and Scallops

Due Monday December 13

Introduction: The Lotka and Volterra exchange

Reference: An extract from Martin Braun's book

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 $b_{11} = b_{22} = 0$.

1. The predator-prey model

In the previous project with studied some of the elementary properties 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*} In the previous project we determined conditions on the coefficients that guarantee that the system has a nontrivial equilibrium $(\bar{x},\bar{y})$, that is, $\bar{x}>0$, $\bar{y}>0$. In this project we assume that those conditions hold, and then we we introduce harvesting proportional to the population of each species \begin{align*} \frac{dx}{dt} &= ( a_1 - b_{11} x - b_{12} y)x - \alpha_1 x, \\ \frac{dy}{dt} &= (-a_2 + b_{21} x - b_{22} y)y - \alpha_2 y. \end{align*} How does harvesting affect the equilibrium $(\bar{x},\bar{y})$? How does that explain the rise in the percentage of sharks in the Adriatic Sea during the World War I?

2. Node or spiral?

In Project #4 you used the coefficients \[ a = \begin{pmatrix} 4 \\ 1 \end{pmatrix}, \quad b = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \] to plot a phase portrait of the predator-prey model. A visual inspection seems to suggest that the equilibrium $E_4$ at $(5/2,3/2)$ is a spiral. Verify, by a linearlization analysis, that $E_4$ is indeed a spiral.

It is possible, however, to have a stable node (not a spiral) $E_4$ in the first quadrant. To demonstrate that, search for suitable coefficients $a_i$ and $b_{ij}$ selected from the set of numbers assigned to you in the following table:

Plot the system's phase portrait within the first quadrant. Note that the plotting ranges xmax and ymax from the previous project will almost certainly be wrong choices here. Apply your judgment to choose a plotting region in the $x$-$y$ plane which best shows the salient aspects of the phase portrait. Plot the nullclines first to get an idea where to focus your attention.

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 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