QUESTION 13. [7 pts] The number of insects in my dead brain is modeled by: It+1 = , It (1 - It) - hat for t = 0, 1, 2, ..., where h is the harvest factor. We need h 2 0 such that the Yield is maximum. a) give the updating function. Answer: f(x) = 2x(1 -x) - hx, yay b) Find Eq points. Some may depend on h. Hint: Solve x = f(x), get r; = 0 and x2 = 3-2h c) for what values of h is there a strictly positive Eq Point? Hint: did you ever solve: 3 - 2h > 0? d) For the strictly positive Eq P found above, say r* , find h such that Y(h) = hx* is maximal. Why is it a global max? Hint: Take the derivative of Y(h) = h3-2h as we did in class, and create a table, to find the LOCAL Max. Justify it is also a Global, looking at the endpoints of the domain. e) Find that max yield. Hint: Plug the above h you found in Y, of course you need a calc .. .. .. f) is the max yield sustainable? In other words: is your eq point stable? Use calculus! Hint: Compute f'(x), then solve If'(3 2h) |