1. The population of the nittany mouse (not a real species) is assumed to grow by 40% each year. Let P = P(t) be the mouse population at time t, in years. (a) Formulate the differential equation modeling the nittany mouse population. Before you proceed to part (b): Did you follow the italicized instructions at the top of the page? Does your differential equation in (a) have a letter in it whose numerical value you know, but did not insert? If so, rewrite the differential equation, substituting that numerical value in place of the letter. Keep that number in place throughout the rest of your work on this problem. (b) Do a qualitative analysis of the differential equation in order to generate a solution portrait for all mathematical solutions, and another portrait for physically viable solutions. (c) Find the general solution to your differential equation. (d) Assuming the current population is 1000, what will the population be in five years? (e) How long will it take for the population to become 10,000? 2. Concerned that the nittany mouse population will get out of control, the Pennsylvania Department of Conservation and Natural Resources (DCNR) introduces the nittany hawk (not a real species), which preys on nittany mice. DCNR estimates that the hawks will eat 500 mice per year. (a) Formulate the differential equation modeling the nittany mouse population in this new scenario. Before you proceed to part (b): Did you follow the italicized instructions at the top of the page? Does your differential equation in (a) have letters in it whose numerical values you know, but did not insert? If so, rewrite the differential equation, substituting those numerical values in place of the letters. Keep those numbers in place throughout the rest of your work on this problem. (b) Do a qualitative analysis of the differential equation in order to generate a solution portrait for all mathematical solutions, and another portrait for physically viable solutions. (c) Does your new model predict that the mice and hawks can co-exist? Explain. (d) Find the general solution to your differential equation. (e) Assuming the current mouse population is 1000, and your model predicts that the mouse popu- lation will grow, what will the mouse population be in five years? If instead your model predicts extinction for the mice, how long will that take? 3. Nittany hawks are dependent on nesting sites that only exist in Pennsylvania. (Play along.) Therefore their population is limited to how many PA can sustain. DCNR estimates this number to be 20,000. Without space limitations the hawks would grow at a constant 20% per year relative growth rate. (a) Formulate the differential equation modeling the nittany hawk population. (This model has nothing to do with problems 1 and 2.) Before you proceed to part (b): Did you follow the italicized instructions at the top of the page? Does your differential equation in (a) have letters in it whose numerical values you know, but did not insert? If so, rewrite the differential equation, substituting those numerical values in place of the letters. Keep those numbers in place throughout the rest of your work on this problem. (b) Do a qualitative analysis of the differential equation in order to generate a solution portrait for physically viable solutions. (c) Suppose there are 5000 nittany hawks now. How many will there be in five years? (d) How long will it take the hawk population to rise to 15,000? 4. DCNR has detected a mysterious disease afflicting the nittany hawks and wiping out about 500 hawks per year. (a) Formulate the differential equation modeling the nittany hawk population in this new scenario (with the same assumptions from problem 3 still in place). Before you proceed to part (b): Did you follow the italicized instructions at the top of the first page? Does your differential equation in (a) have letters in it whose numerical values you know, but did not insert? If so, reurite the differential equation, substituting those numerical values in place of the letters. Keep those numbers in place throughout the rest of your work on this problem. (b) Do a qualitative analysis of the differential equation in order to generate a solution portrait for physically viable solutions. (c) As long as the disease still persists, and there are 5000 hawks now, does the model predict extinction for the hawk? Explain. (d) What is the new effective capacity Jevel for the hawk population? Explain.