1. The population of frogs in a bog can be modeled by the logistic equation % = 0061' 105132 = 105P{6000 P) where P(t) is the number of frogs in the bog after it years. Suppose that, at time t = 0, there are 1000 frogs in the bog. (a) Use qualitative analysis to sketch the graph of the frog population over time. (b) Use separation of variables to solve the initial value problem.(1) (Hint: You'll need to use partial fractions.) (c) Use a calculator or computer to graph the solution you found in (b); make sure that it matches your sketch in (a). (d) Does the frog population ever reach 4000 frogs? If so, at what time? (Note: One of these questions can be answered just using qualitative analysis; the other cannot.) 2. For each of the differential equations below, use qualitative analysis to sketch a family of representative solutions. Identify the equilibrium solutions of each differential equation, and say whether they are sta ble or unstable. Then, in another color pen or pencil, sketch the graphs of the solutions corresponding to the given initial conditions. (a) E = 9'? - 4. 9(0) -1, 9(0) = (b) g = (P 2H7 P); 1303) = 1, 13(0) = 3. 3. The sh and game department in a certain state is planning to issue hunting permits to control the deer population (one deer per permit). It is known that if the deer population falls below a certain level In, the deer will become extinct. It is also known that if the deer population rises above the carrying capacity M, the population will decrease back to M through disease and malnutrition. (a) Use qualitative analysis to sketch a family of representative solutions of the following model for the growth rate of the deer population as a function of time: tip 3 = rP(M P)(P m), where P is the population of the deer and \"r is a positive constant of proportionality. dP (b) Briey explain how this model differs from the logistic model: E = TP(M P)