Need help solving the member 1 alternative model, for a=1 only. That a=1 then is used in part 4 question 1. So only the highlighted portions need to be done, thank you.
-.. .. 1. '.1-r.-..-. ...,,-= 1. r ..-r-.'. -m -.- 'r-Ivv'vr .-r 1.-. .r1...- '-.n-1-. 7-1 -. 7.'-.... ~ 1 18. How Many Deer Are There? The numberof deer in a state park was 1,000 six years ago and today it is 2,003. 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 l0,000 deer in the park, overgrazing may change the character of the park andr'or Imp numbers ofdeer may starve in the winter.) Your group has been hired to assist the park service in predicting both long and short term population trends. Pant Your rst task is to investigate a family ofpossible models forthe population ofthe deer. Call PU} the number ofdeer in the park i' years aer 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 Zyears from today. IL A general rule for forecasting the population of the deer for any time in the future. 5. The time (ifever) when the population will reach 10,030. 6. lithe deer population follows this model any problems or advantages for the park service. Model I it Find the constant rate of chan for the deer population which gives a solution that ts 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 (niembm#} 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 at. These models can be described by the differential equation dedr 2 kl\" or P' = kP\". Member 1 will investigaterr = .2, l, 2 Member 2 will investigate rr = .3, l, 3 Member 3 will investigate rr = .4. l. 4 Member 4 will investigate rr = .5, 1,5 The group should now have investigations of at least 10 different models. Analyze the models and discuss the following problems as well as any other interesting facts and properties of the models: Which modelsI: if any) keep the population under 10.000 in the next 5 years? Which models {if any) will give a model which keeps the population under 10,610 in the next nyears? Which models (if any} will give a mode] which keeps the population under 10,[Ill forever? 18. HourI Mary Deerme There? 2131r Part2 After you have completed your analysis of Part 1, the park service understarrdsthe models better but is tryingto decide which model describes the population of the deer the best. In an old le they have found that the deer population was EEK] eight years ago, 600 eleven years ago and 500 fteen years ago. Using this inflammation 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. Ifyou can nd a \"better\" modal than any of the models, include it in your report as well as explaining why it is better. NOTE: Do not use theerrtra data given in Part 2 when you work Parts 3 and 4. Part3 In order to keep the deer population in check the park service is considering issuing deer hunting licenses. Based upon pastexperience, for every 100 licenses issued 60 deer will he killed. Modify only the a = 1 mode] to take into account issuing 5m licenses peryear. 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 he in the future? 3. When (if ever) will the population reach 10,000? 4-. When (if ever) will the population fall beneath 500? 5. Carl 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 ofthe population to 1500 and then keep the population at about 1500 thereafter: Ifthis is possible tell them how to do it by controlling the nrunber of hunti ng licenses. Part 4 A different method of population control is to introduce a predator: If the forest service introduces wolves, (you may assume wolyes were extinct in this area in the post), for every 20 wolves, Ell] deer per year will die. The EWiDB is considering introducinga pack of lwolves. As long a the deer population is ll]I times the wolfpopulation, the wolf populationwill grow at the rate of 10% per year, if the deer population is between 5 and 10 times the wolfpopulation, then the wolf population will grow at the rate of 5% per year; if ttre deer population is between 2 and 5 titres the wolf populationJhe wolf population will grow at 1% per year; and if the deer population is less than twice tie wohr population, then the wolfpcpulationwill decrease at 2% per year. Usingthe model a = l (with no hunting licenses) project what: the introduction ofwolves 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 neth 5 years and forthe next 10 years
