There is considerable evidence to support the theory that for some species there is a minimum population m such that the species will become extinct if the size of the population falls below m. This condition can be incorporated into the logistic equation by introducing the factor s1 2 myPd. Thus the modified logistic model is given by the differential equation dP dt kPS1 2 P M DS1 2 m P D (a) Use the differential equation to show that any solution is increasing if m , P , M and decreasing if 0 , P , m. (b) For the case where k 0.08, M 1000, and m 200, draw a direction field and use it to sketch several solution curves. Describe what happens to the population for various initial populations. What are the equilibrium solutions? (c) Solve the differential equation explicitly, either by using partial fractions or with a computer algebra system. Use the initial population P0. (d) Use the solution in part (c) to show that if P0 , m, then the species will become extinct. [Hint: Show that the numerator in your expression for Pstd is 0 for some value of t.]

1 25 Multiple Choice & Essay on Chapters 3, 13, 14, & 15. Chapter 3: American Federalism 3.1 Defining Federalism Federalism ● A constitutional arrangement in which power is distributed between a central government and states. The national and states exercise direct authority over individuals. ● The Constitution divides governmental powers between the national government and state governments, giving clearly defined functions to each. ● Nearly 90,000 governments in the U.S., including 1 federal government and 50 state government