Math 152H, Fall 2008Calculus IIProject Three: A Curious Sequence1 OverviewIn this project we study sequences defined by the recurrence relationan+1= kan(1 − an). (1)Different choices of the parameter k and of the first term a0result in different sequences. Thissequence has been used by ecologists to model the evolution of insect populations. In sucha model, there is assumed to be a maximum population size (the ”carrying capacity”) thatcan be supported by the environment. The n-th term in the sequence, an, then representsthe fraction of the maximum population that is alive in the n-th generation. When anisclose to zero, 1 − an≈ 1 and so e quation (1) is approximately given by an+1≈ kan. In thiscase the population grows exponentially. However as anincreases towards 1 the factor 1 − angets closer to 0 which causes the population to decrease. In summary, the factor kanmodelsthe grow of population due to births (and deaths) when the population is not limited by theavailable resources and the factor 1 − anis included to account for the finite resources (andhence finite carrying capacity) of the environment.The goal of the project is to study the diffe rent sorts of long term behaviour these se-quences exhibit. Such an analysis helps ecologists answer questions like: Will the populationstabilize to a limiting value? Will it change in a cyclical fashion? Under what circumstancesdoes the population vary in a random or chaotic manner?To study the properties of (1) it is useful to introduce the functionf(x) = kx(1 − x). (2)Then an+1= f(an).2 Questions1. For ecology applications we need 0 ≤ an≤ 1 for all n. Suppose that 0 ≤ a0≤ 1. Usecalculus 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 andgraph the first N terms in the sequence (1). The input parameters to your programshould be N, k and a0. The output should be the table of values (n, an) together withthe graph of anversus n . Note: In Matlab, to plot a sequence anversus n store thenumbers n and anin arrays n and a and use the command plot(n,a,’bx’). This will put1a 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 papercalculation of the first few terms. Then use your program to answer the followingquestions.3. Calculate about 25 terms of the sequence with a0= 0.5 for two values of k such that1 < k < 3. Graph the sequences. Do they appear to converge? If so, what do youthink the limits are? Rep eat for a different choice of a0with 0 < a0< 1. Does thelimit depend on the choice of a0or k or both?4. Suppose L = limn→∞anexists. Do an algebraic calculation to determine the possiblevalues 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 yourplot graph the points (a0, a1), (a1, a2), (a2, a3),.... Join each point (an, an+1) to the nextpoint with a line. (You can modify your matlab function to do this.) What happensto the points (an, an+1) as n → ∞?6. In general prove that if the sequences converges then the points (an, an+1) converge toan 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. Whatdo 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 behaviourof the sequence. What happens if you change a0by 0.001? This type of behaviour iscalled 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 theexamples above we are now going to study how the sequence depends on the first terma0. For k = 1, 3, 4 and n = 1, 2, 3, 4, 5 plot anversus a0where 0 ≤ a0≤ 1. Another wayto think about what I am asking you to do here is the following. For a fixed value of kwe can think if a1as 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 UpFor your write up please include all calculations, any Matlab code you write, output and plotsgenerated by Matlab, and a brief introduction and conclusion showing that you understandthe point of the project.Some additional points to keep in mind are:21. The write up should be a self-contained, i.e., it should make sense even if you don’thave this Project Assignment Sheet.2. Your audience is a fellow class mate who has studied the same material you have inclass, but has not done this project.3. Don’t talk about your experiences doing the project. Rather, summarize your scien-tific/mathematical findings/results.4. Your introduction should be in the form of an ”executive summary” of the results youfound, rather than simply being a restatement of the problem. For this be as conciseand 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 eachfigure.7. Do not assume that the author of this Project Assignment Sheet is infallible. I maywell have made a