Part 2 After you have completed your analysis of Part 1, the park service understands the models better but is trying to decide which model describes the population of the deer the best. In an old file they have found that the deer population was 800 eight years ago, 600 eleven years ago and 500 fifteen years ago. Using this information decide which of your models from Part 1 best describes the population of deer. An important part of your report should be a description of your method for determining which was best as well as showing how you applied it to each model. If you can find a "better" model than any of the models, include it in your report as well as explaining why it is better. NOTE: Do not use the extra data given in Part 2 when you work Parts 3 and 4. Part 3 In order to keep the deer population in check the park service is considering issuing deer hunting licenses. Based upon past experience, for every 100 licenses issued 60 deer will be killed. Modify only the a D 1 model to take into account issuing 500 licenses per year. You may assume that there was no deer hunting in the past. 1. What differential equation now models the deer population? 2. According to this model what will the population be in the future? 3. When (if ever) will the population reach 10,000? 4. When (if ever) will the population fall beneath 500? 5. Can the forest service keep the deer population stable by hunting? If this is possible describe how they can do it. 6. Can the forest service reduce the deer population by hunting? The service would like to have a gradual reduction of the population to 1500 and then keep the population at about 1500 thereafter. If this is possible tell them how to do it by controlling the number of hunting licenses. Part 4 A different method of population control is to introduce a predator. If the forest service introduces wolves, (you may assume wolves were extinct in this area in the past), for every 20 wolves, 80 deer per year will die. The service is considering introducing a pack of 100 wolves. As long as the deer population is 10 times the wolf population, the wolf population will grow at the rate of 10% per year, if the deer population is between 5 and 10 times the wolf population, then the wolf population will grow at the rate of 5% per year; if the deer population is between 2 and 5 times the wolf population, the wolf population will grow at 1% per year; and if the deer population is less than twice the wolf population, then the wolf population will decrease at 2% per year. Using the model a D 1 (with no hunting licenses) project what the introduction of wolves will do to the deer population 1. for the next 5 years, 2. for the next 10 years. 3. What would happen in the long run? 4. Include projected population data for both wolves and deer for the next 5 years and for the next 10 years. O O2:36 9 46 How man...r are there D . .. 18. How Many Deer Are There? The number of deer in a state park was 1,000 six years ago and today it is 2,000. Based upon data from other habitats, the park service estimates that no more than 10,000 deer can inhabit the park. (If there are more than 10,000 deer in the park, overgrazing may change the character of the park and/or large numbers of deer may starve in the winter.) Your group has been hired to assist the park service in predicting both long and short term population trends. Part 1 Your first task is to investigate a family of possible models for the population of the deer, Call P (t ) the number of deer in the park t years after some starting time. Your group should decide what the starting time is and use it throughout your report. The group must also agree on each member's number since this will be needed to investigate the models Your report should include the following information for each model: 1. A verbal description. 2. A differential equation including initial conditions. 3. The deer population 1 year from today, the deer population 5 years from today and the deer population 20 years from today. 4. A general rule for forecasting the population of the deer for any time in the future. 5. The time (if ever) when the population will reach 10,000. 6. If the deer population follows this model any problems or advantages for the park service. Model 1 a. Find the constant rate of change for the deer population which gives a solution that its the two given population counts exactly. Use this model for the population from 6 years ago until today. b. Using only the information that the population today is 2,000 and that the change in the population of deer is (members) x 100 deer per year every year from now on, predict the population starting today. So you should end up with a piecewise function for the deer population starting 6 years ago. Alternative Models Each member of your group will investigate these models for selected values of a. These models can be described by the differential equation dp / at = kPa orP' = kPa The (ais above are exponents Member 1 will investigate a = :2; 1; 2 Member 2 will investigate a = :3; 1; 3 Member 3 will investigate a = :4: 1;4 Member 4 will investigate a = :5; 1; 5 The group should now have investigations of at least 10 diff Analy models and discuss the following problems as well as any oth. Ming facts properties of the models: O O