Project 5, MATH 481, Spring 2008
Population dynamics: Competing species

Assignment of 2008–04–07

Introduce the following model for populations x(t) and y(t) that compete for common resources:

dx/dt = ( a₁ − b₁ x − c₁ y ) x,
dy/dt = ( a₂ − b₂ x − c₂ y ) y.

Case 1:

a₁ / b₁ > a₂ / b₂ and a₂ / c₂ > a₁ / c₁

Case 2:

a₁ / b₁ < a₂ / b₂ and a₂ / c₂ < a₁ / c₁

Case 3:

a₁ / b₁ > a₂ / b₂ and a₂ / c₂ < a₁ / c₁

Case 4:

a₁ / b₁ < a₂ / b₂ and a₂ / c₂ > a₁ / c₁

In cases 1 and 2, study local behavior near equilibria. Illustrate your conclusions by assigning values to the coefficients and sketching representative phase portraits. Cases 3 and 4 are easier. Analyze them if you want to.

Note added 2008–04–08

In the earlier version of this web page, the right hand side of the second differential equation was garbled. I have fixed that now.

Due date

This assignment is due on Wednesday April 16.