Question

Exercise 1.6.2: Draw the phase diagram for different possibilities. Note that these possibilities are A> B, or A B orA and B both complex (i.e. no real soluins). Hint: Fix some simple k and M and then vary h For example, let M-8 and k-0.1. When h-1, then A and B are distinct and positive. The graph we will get is given in Figure 1.10 on the next page. As long as the population starts above B which is approximately 1.55 million, then the population will not die out. It will in fact tend towards A6.45 million. If ever some catastrophe happens and the population drops below B, humans will die out, and the fast food restaurant serving them will go out of business. When h-1.6, then A B-4. There is only one critical point and it is unstable. When the population starts above 4 million it will tend towards 4 million. If it ever drops below 4 llion humans will die out on the planet. This scenario is not one that we (as the human fast food proprietor) want to be in. A small perturbation of the eilibrium state and we are out of business. There is no room for error. See Figure 1.11 on the following page Finally if we are harvesting at 2 on humans per year, there are no critical points. The population will always plummet towards zero, no matter how well stocked the planet starts. See Figure 1.12 on the next page

*Differential Equations*

Answer 1

Given data, A > B, (or) A = B (or) A and B Both compare for Example let m = 8 & k = 0.1 when h = 1 then A and B are distinct and positive. A = kM + squareroot(kM)^2 - 4hk/2k B = kM - squareroot (kM)^2 - 4hk/2k As long or the population state above B which is 1.5 million, then the population will not die out. It will input and towards A 6.45 million. If our some happens and the population drops below B, human will die chat, and the test toad saving often will go out of business. \r\n rightarrow when h = 1.6 then A = B = 4, there is only one critical present and it is unstable when population starts above 4 million it will tend forwards 4 million It will drops below 4 million humans will die out on the planet. This is not one that we as the human that load properties want to be in.