A solution to the Wright-Fisher model of genetic drift.

COURSE: MATH 5900

TIME: Arranged

PLACE: Arranged (Math Lab likely)

INSTRUCTOR: Dr. Robert Gardner, OFFICE: Room 308F of Gilbreath Hall

OFFICE HOURS: By appointment

PHONE: 439-6979, Math Office 439-4349

E-MAIL: gardnerr@etsu.edu .

HOMEPAGE: faculty.etsu.edu/gardnerr/gardner.htm .

CLASS WEBPAGE: A webpage containing this syllabus, classnotes, and homework assignments is available at faculty.etsu.edu/gardnerr/mathbio/sillsum2010.html.

PRIMARY TEXT: Evolutionary Games and Population Dynamics, by J. Hofbauer and K. Sigmund, Cambridge University Press, 1998. Copies of chapters of this book will be provided.
ADDITIONAL TEXT: An Introduction to Chaotic Dynamical Systems, 2nd Editions, Addison-Wesley, 1989. Copies of chapters of this book will be provided.
ADDITIONAL REFERENCES: I have put together a few PDF files relevant to our class:

• Analysis of the Logistic Recurrence Relation for 1 < λ < 3, λ ≠ 2.
• Emprical Test of Lotka-Volterra.
• Analysis of Fixed Points (i.e. Critical Points) in a Two Dimensional Nonlinear System.
• A Mathematician Looks at Chaos.

GRADING: Your grade will be determined based on your performance on the assigned homework problems and projects. Grades will be assigned based on a 10 point scale, with plus and minus grades being given as appropriate.

TENTATIVE OUTLINE: We will try to adhere to the following ambitious outline.

• Week 1: The Wright-Fisher Equation. Source: The Wright-Fisher Model, ETSU M.S. Thesis by Malissa Brown, 1995.
• Week 2: Logistic Growth. Source: Chapter 1 of Hofbauer and Sigmund.
• Week 3: Lotka-Volterra Equations for Predator-Prey Systems. Source: Chapter 2 of Hofbauer and Sigmund.
• Week 4: The Lotka-Volterra Equation for Two Competing Species, Ecological Equations for Two Species. Source: Chapters 3 and 4 of Hofbauer and Sigmund.
• Week 5: Discrete Dynamical Systems as Population Genetics. Source: Chapter 18 of Hofbauer and Sigmund.

HOMEWORK ASSIGNMENTS
WEEKS 1: Find a way with MatLab or (preferably) Maple to numerically solve equation (2.6) on page 12 of The Wright Fisher Model. Prepare graphs of the solution for different initial conditions, similar to what is given pictorially at the top of this webpage.
WEEK 2: Hofbauer and Sigmund 1.3.1, 1.5.1 for R=2, 1.5.2. Due Monday 6/21.
WEEK 3: Devaney 1.12.3, 1.12.4 (see below). Due Friday 6/25.
WEEK 4: Devaney 3.4.3, 3.4.4. Due Friday 7/9.
WEEK 5: Devaney 4.3.1. Due Friday 7/9

PROBLEMS
Devaney 1.12.3: Let λ < 0. Prove that E_λ(x) has a unique fixed point that is attracting if λ > -e and repelling if λ < -e.
Devaney 1.12.4: Prove that E_λ² is concave up for x such that E_λ'(x) > -1, and concave down if E_λ'(x) < -1.

Charles Darwin (1809-1882)

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Last updated: July 5, 2010.