# MathBio HW6, due Wed April 16

1. A cooperation model.

Consider the following two-species model. $\displaystyle \dot x = x(1 - x) + a x y$ $\displaystyle \dot y = c y (1 - y) + c b xy$

where $a$, $b$ and $c$ are positive constants.

1. Under what circumstances is there a positive coexistence equilibrium?
2. In the case where there exists a positive equilibrium, find ALL equilibria, determine their stability and sketch the phase plane.
3. What does the phase plane look like when there is no positive coexistence state?

2. On the half life of one-hit wonders

Suppose that there is a band that puts out a great first album, but then they soon run out of ideas.  The first album has 15 songs, of which 5 are good. The second album has 12 songs, of which 3 are good. The last album has 10 songs, of which only 1 is good.  Use Bayes’ Theorem to answer the following questions.

1. Suppose that a radio station picks songs at random to play and you hear one of their good songs on the radio.  What is the probability that song is from the first album?
2. Suppose that another radio station plays only good songs and chooses from them at random.  You hear one of the band’s good songs on this station.  What is the probability the song is from the first album?