ENDY 6013 - ENVIRONMENTAL DYNAMICS

HOMEWORK ASSIGNMENT #7 - SENSITIVITY TRAINING

DUE FRIDAY, 15 November 2002

Earlier this week there was some discussion (though short-lived) regarding the nature of knowledge, the relevance of our understanding, and the degree to which acquired knowledge of past behavior of processes or systems could be used to predict the future state of those processes or systems. For your assignment this week, you will explore a simple dynamical system in an attempt to determine whether or not you know what you think you know.

You will be asked to utilize a spreadsheet to make predictions about a simple mathematical function that was devised in the middle of the last century to describe the behavior of populations. The equation we will use is called the Logistic Difference Equation, and you can read a very short but highly informative tutorial on this equation by following the hyperlink.

The logistic difference equation describes a simple feedback system. It assumes a starting point (let's call it x) that initializes the system. This initial value can be any number other than zero between 0 and 1. In most applications, this number represents a proportion of a population (i.e.an initial value 0.5 equates to 50% of the population). The equation is iterated by multiplying the initial value by a growth parameter, usually represented as r. The exponential growth of x is prevented from running away by inserting a term that limits x if it becomes too large. Thus, the full form of the logistic equation is

x_n = rx(1-x)

Program the equation above in Excel so that you can explore its behavior using the given parameters and questions below.

1. Create a spreadsheet such that this equation iterates at least 100 times. Make the spreadsheet model functional by providing a means to alter the initial x value and the growth parameter, r. Once you are sure you have programmed the equation properly, create a chart to plot iteration number versus x, and alter the inputs as described below and attempt to describe your observations of the behavior of this system.

2. Enter a value of 0.5 for the initial value of x and 1.0 for r. What happens to your graph? Can you describe why this occurs? (Hint: think in terms of populations and population dynamics).

3. Enter a value of 0.5 for the initial value of x and 2.0 for r. Now what happens to your graph? Can you describe why this occurs?

4. Change the initial value of x to 0.0001 but leave r = 2.0. Now what happens to your graph? Can you describe why this occurs?

5. Leave the initial value of x = 0.0001 but change r to 2.7. Now what happens to your graph? Can you describe why this occurs? How would you describe this particular system after its 46th iteration?

6. Increase r to 2.99. What appears to happen to your curve? How would you describe this system?

7. Increase r to 3.2. What appears to happen to your curve? How would you describe this system?

8. Change x to 0.00000001; r = 3.2. What appears to happen to your curve compared to the previous setting? How would you describe this system?

9. Change x to 0.1 and r = 3.8. What appears to happen to your curve compared to the previous setting? How would you describe this system?

10. Create another column of data in your spreadsheet and set x = 0.100000000001 and r = 3.8. Plot the results of this calculation on the same graph as the previous iteration. What do you observe? How would you describe these systems? Are the two plots similar to each other? Since you already had results from question 9 plotted, were you able to predict the behavior of the new plot from those data? How similar were the input data for each plot?