If at all possible, please work in groups of 2-3 people!

1. Do problem 5 on page 44. This is to find a differential equation which models a population of microorganisms with plenty of food, which buds off a copy of itself about four times a day. Assume the average lifespan is one day. Find a model for the population, and find the solution given that there are 1,000 individuals at time t=0.

2. Suppose we use the same model organism as in problem 1, but the given reproduction rate is only valid where there are very few individuals. Suppose you observe that in your particular dish where you are growing them, there is a stable population of about 5,000 individuals. Modify the model you made to take this into account.

3. Suppose we model a second hypothetical environment.  Colonies grown in this environment sometimes level out at 10,000 individuals, but sometimes the population stabilizes at 20,000 individuals.  We also observe that when we remove (or add) a few individuals from (or to) these stable populations, it takes 10 times as long for the population near 20,000 to recover to the stable state of 20,000 as it does for the a population near 10,000 to recover. Make a model for population growth (or shrinkage) in this environment.  [According to Dr. Uhlenbeck, it is relatively easy to get a model with two unknown constants, but, in fact, you can write down a model with just one unknown constant.]