Cobweb diagram - Cobweb Diagrams With Excel David Gurney...

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Cobweb Diagrams With Excel David Gurney Department of Mathematics Southeastern Louisiana University SLU 10541, 500 Western Ave Hammond, LA 70402 [email protected] October 27, 2004 Abstract Iterative functions are often used to simulate the dynamics of human and animal populations. Unfortunately, if the iterative function is no more complicated than a two-term quadratic, trying to predict the results of repeated application of the function is very difficult. Even studying the numerical results of applying the function again and again may not provide much insight into the long-term behavior of the populations. Cobweb diagrams offer a way around this dilemma. Originally devised for use with rulers and graph paper, cobweb diagrams provide a relatively quick way of representing the repeated application of an iterative function. This talk will demonstrate how to create cobweb diagrams using a Microsoft Excel spreadsheet. The strong and weak points of this method of creating cobweb diagrams, as opposed to using the TI-83 graphing calculator or software programs like Maple and Mathematica, will also be discussed.
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Cobweb Diagrams with Excel 12/16/2004 1 Every once in a while I am given the opportunity to teach a population modeling course at Southeastern Louisiana University. Usually the students who sign up for the course are those who need an extra couple of credit hours in something and their background is not much more than first semester calculus. Therefore I cannot assume the students know much about using Mathematica or Maple or that they even have a certain type of calculator. I usually show students how to use Excel to develop the simple models and also show them how to use Mathematica to help with the analysis of the resulting models. This summer, however, I decided to restrict myself to using Excel for everything we needed to do. You can do quite a bit of population modeling with just Excel. Time series graphs are easy. Phase plane graphs are not much more difficult. Even putting in randomized birth and death rates is fairly simple. But until this year, I had not been able to construct a cobweb diagram in Excel in spite of some hours trying. I was lucky enough to discover the method described here, however, in the few weeks after spring finals and before my population modeling course began in the summer session. Before hand-held calculators and personal computers, cobweb diagrams were a means to avoid some very messy calculations and still demonstrate the result of successive iteration of functions. Even with calculators and computers, the algebraic result of applying a function as simple as a binomial quadratic to itself over and over again can be very complicated after just a few iterations. The example I will use for this paper is the function ( ) 3.57 (1 ) fx x x =− with initial value 0.5. Population modelers call this the logistic function and 3.57 is the logistic constant. The table at right shows the numerical result of twenty iterations of this function. Figure A on the next page shows a plot of the values for each iteration.
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This note was uploaded on 11/20/2010 for the course ECONOMICS 331 taught by Professor Mj during the Fall '10 term at University of Alberta.

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Cobweb diagram - Cobweb Diagrams With Excel David Gurney...

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