Problem 27:
    Jamie Howard
    Randy Keeling
    Jeremy Phillips

Suppose that the number of trees in a certain forest increases at a rate of 10% per year, that the carrying capacity of the forest is 1000 trees per acre and that 10 trees per acre are harvested from the forest every year.

If we let y(t) denote the number of trees per acre at time t, then

(1)

We can determine the equilibrium solutions as the solutions to,

(2)

We can expand this to yield,

(3)

In order to solve this equation, we must apply the quadratic formula,

(4)

and this gives the equilibria to be

(5)

We now determine the vector field by testing f(y) at points in the intervals, 

Thus, the vector field appears to be,

Thus, y=887.30 trees per acre is a stable equilibrium. This means, that after a long period of time, we can expect about 887.30 trees per acre on the land from which the harvesting is being conducted. This means also that 112.70 is unstable. This means that as long as there are more than 112.70 trees per acre, the forest will be able to recover to its stable equilibrium of 887.30 trees.

However, if the number of trees per acre ever drops below 112.70, then the number of trees will decrease rapidly. Thus, harvesting to below this level will cause the forest to die out.