Stochastic population growth: Many populations are appropriately modeled by an exponential function, at least for a limited period of time. But there are many factors contributing to the growth of any...

Stochastic population growth: Many populations are appropriately modeled by an exponential function, at least for a limited period of time. But there are many factors contributing to the growth of any population, and many of them depend on chance. There are a number of ways to produce stochastic models. We consider one that is an illustration of a simple Monte Carlo method. We want to model a population that is initially 500 and grows at an average rate of 2% per year. To do this we make a table of values for population according to the following procedure. The first entry in the table is for time t = 0, and it records the initial value 500. To get the entry corresponding to t = 1, we roll a die and change the population according to the face that appears, using the following rule.

To get the entry corresponding to t = 2, we roll a die and change the population from t = 1, again using the above rule. This procedure is then followed for t = 3, and so on.

a. Using this procedure, make a table recording the population values for years 0 through 10.

b. Plot the data points from your table and the exponential model on the same screen. Comment on the level of agreement.