Math 152H Fall 2008 Calculus II Project Three A Curious Sequence 1 Overview In this project we study sequences defined by the recurrence relation an 1 kan 1 an 1 Different choices of the parameter k and of the first term a0 result in different sequences This sequence has been used by ecologists to model the evolution of insect populations In such a model there is assumed to be a maximum population size the carrying capacity that can be supported by the environment The n th term in the sequence an then represents the fraction of the maximum population that is alive in the n th generation When an is close to zero 1 an 1 and so equation 1 is approximately given by an 1 kan In this case the population grows exponentially However as an increases towards 1 the factor 1 an gets closer to 0 which causes the population to decrease In summary the factor kan models the grow of population due to births and deaths when the population is not limited by the available resources and the factor 1 an is included to account for the finite resources and hence finite carrying capacity of the environment The goal of the project is to study the different sorts of long term behaviour these sequences exhibit Such an analysis helps ecologists answer questions like Will the population stabilize to a limiting value Will it change in a cyclical fashion Under what circumstances does the population vary in a random or chaotic manner To study the properties of 1 it is useful to introduce the function f x kx 1 x 2 Then an 1 f an 2 Questions 1 For ecology applications we need 0 an 1 for all n Suppose that 0 a0 1 Use calculus to find the values of k for which 0 an 1 for all positive integers n 2 Write a computer program preferably in Matlab to calculate a table of values and graph the first N terms in the sequence 1 The input parameters to your program should be N k and a0 The output should be the table of values n an together with the graph of an versus n Note In Matlab to plot a sequence an versus n store the numbers n and an in arrays n and a and use the command plot n a bx This will put 1 a blue cross at each point n a n Other colors are red r black k green g etc Check your program works correctly by comparing its output to a pencil and paper calculation of the first few terms Then use your program to answer the following questions 3 Calculate about 25 terms of the sequence with a0 0 5 for two values of k such that 1 k 3 Graph the sequences Do they appear to converge If so what do you think the limits are Repeat for a different choice of a0 with 0 a0 1 Does the limit depend on the choice of a0 or k or both 4 Suppose L lim an exists Do an algebraic calculation to determine the possible n values of L Compare your result to the graphs in the previous question 5 For one of the k values you chose above plot the function in equation 2 On your plot graph the points a0 a1 a1 a2 a2 a3 Join each point an an 1 to the next point with a line You can modify your matlab function to do this What happens to the points an an 1 as n 6 In general prove that if the sequences converges then the points an an 1 converge to an intersection point of the line y x with the graph of 2 7 Now we are going to examine some different sorts of behaviour for different values of k Calculate and graph terms in the sequence for a value of k between 3 and 3 4 What do you notice 8 Experiment with values of k between 3 4 and 3 5 What happens to these sequences 9 For values of k between 3 6 and 4 plot at least 100 terms and comment on the behaviour of the sequence What happens if you change a0 by 0 001 This type of behaviour is called chaotic and is exhibited by insect populations under certain conditions 10 To get more of a feeling for the different sorts of behaviour we have observed in the examples above we are now going to study how the sequence depends on the first term a0 For k 1 3 4 and n 1 2 3 4 5 plot an versus a0 where 0 a0 1 Another way to think about what I am asking you to do here is the following For a fixed value of k we can think if a1 as being a function of a0 In fact this is just the function a1 f a0 where f is given in 2 Similarly a2 f f a0 involves composition of f with itself and so on What do you learn from these graphs 3 The Write Up For your write up please include all calculations any Matlab code you write output and plots generated by Matlab and a brief introduction and conclusion showing that you understand the point of the project Some additional points to keep in mind are 2 1 The write up should be a self contained i e it should make sense even if you don t have this Project Assignment Sheet 2 Your audience is a fellow class mate who has studied the same material you have in class but has not done this project 3 Don t talk about your experiences doing the project Rather summarize your scientific mathematical findings results 4 Your introduction should be in the form of an executive summary of the results you found rather than simply being a restatement of the problem For this be as concise and precise as you can 5 Be as precise as you can 6 Make sure your figures have legends Also comment on what you learn from each figure 7 Do not assume that the author of this Project Assignment Sheet is infallible I may well have made a mistake 3