{"id":92,"date":"2021-03-25T02:20:58","date_gmt":"2021-03-25T02:20:58","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/the-logistic-equation-2\/"},"modified":"2021-12-09T02:54:55","modified_gmt":"2021-12-09T02:54:55","slug":"the-logistic-equation-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/the-logistic-equation-2\/","title":{"raw":"Problem Set: The Logistic Equation","rendered":"Problem Set: The Logistic Equation"},"content":{"raw":"

For the following problems, consider the logistic equation in the form [latex]P\\prime =CP-{P}^{2}[\/latex]. Draw the directional field and find the stability of the equilibria.<\/p>\r\n\r\n

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1.\u00a0<\/strong>[latex]C=3[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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2.\u00a0<\/strong>[latex]C=0[\/latex]<\/p>\r\n[reveal-answer q=\"6533\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"6533\"]\"A[latex]P=0[\/latex] semi-stable[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>[latex]C=-3[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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4.\u00a0<\/strong>Solve the logistic equation for [latex]C=10[\/latex] and an initial condition of [latex]P\\left(0\\right)=2[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"735327\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"735327\"][latex]P=\\frac{10{e}^{10x}}{{e}^{10x}+4}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>Solve the logistic equation for [latex]C=-10[\/latex] and an initial condition of [latex]P\\left(0\\right)=2[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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6.\u00a0<\/strong>A population of deer inside a park has a carrying capacity of [latex]200[\/latex] and a growth rate of [latex]2\\text{%}[\/latex]. If the initial population is [latex]50[\/latex] deer, what is the population of deer at any given time?<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"765811\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"765811\"][latex]P\\left(t\\right)=\\frac{10000{e}^{0.02t}}{150+50{e}^{0.02t}}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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7.\u00a0<\/strong>A population of frogs in a pond has a growth rate of [latex]5\\text{%}[\/latex]. If the initial population is [latex]1000[\/latex] frogs and the carrying capacity is [latex]6000[\/latex], what is the population of frogs at any given time?<\/div>\r\n<\/div>\r\n<\/div>\r\n
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8. [T]<\/strong> Bacteria grow at a rate of [latex]20\\text{%}[\/latex] per hour in a petri dish. If there is initially one bacterium and a carrying capacity of [latex]1[\/latex] million cells, how long does it take to reach [latex]500,000[\/latex] cells?<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"667313\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"667313\"][latex]69[\/latex] hours [latex]5[\/latex] minutes[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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9. [T]<\/strong> Rabbits in a park have an initial population of [latex]10[\/latex] and grow at a rate of [latex]4\\text{%}[\/latex] per year. If the carrying capacity is [latex]500[\/latex], at what time does the population reach [latex]100[\/latex] rabbits?<\/div>\r\n<\/div>\r\n<\/div>\r\n
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10. [T]<\/strong> Two monkeys are placed on an island. After [latex]5[\/latex] years, there are [latex]8[\/latex] monkeys, and the estimated carrying capacity is [latex]25[\/latex] monkeys. When does the population of monkeys reach [latex]16[\/latex] monkeys?<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"707669\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"707669\"][latex]8[\/latex] years [latex]11[\/latex] months[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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11. [T]<\/strong> A butterfly sanctuary is built that can hold [latex]2000[\/latex] butterflies, and [latex]400[\/latex] butterflies are initially moved in. If after [latex]2[\/latex] months there are now [latex]800[\/latex] butterflies, when does the population get to [latex]1500[\/latex] butterflies?<\/div>\r\n<\/div>\r\n<\/div>\r\n

The following problems consider the logistic equation with an added term for depletion, either through death or emigration.<\/p>\r\n\r\n

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12. [T]<\/strong> The population of trout in a pond is given by [latex]P\\prime =0.4P\\left(1-\\frac{P}{10000}\\right)-400[\/latex], where [latex]400[\/latex] trout are caught per year. Use your calculator or computer software to draw a directional field and draw a few sample solutions. What do you expect for the behavior?<\/p>\r\n[reveal-answer q=\"200084\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"200084\"]\"A[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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13.\u00a0<\/strong>In the preceding problem, what are the stabilities of the equilibria [latex]0<{P}_{1}<{P}_{2}?[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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14. [T]<\/strong> For the preceding problem, use software to generate a directional field for the value [latex]f=400[\/latex]. What are the stabilities of the equilibria?<\/p>\r\n[reveal-answer q=\"976439\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"976439\"]\"A[latex]{P}_{1}[\/latex] semi-stable[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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15. [T]<\/strong> For the preceding problems, use software to generate a directional field for the value [latex]f=600[\/latex]. What are the stabilities of the equilibria?<\/div>\r\n<\/div>\r\n<\/div>\r\n
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16. [T]<\/strong> For the preceding problems, consider the case where a certain number of fish are added to the pond, or [latex]f=-200[\/latex]. What are the nonnegative equilibria and their stabilities?<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"959523\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"959523\"]\"A<\/span>\r\n[latex]{P}_{2}>0[\/latex] stable[\/hidden-answer]<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

It is more likely that the amount of fishing is governed by the current number of fish present, so instead of a constant number of fish being caught, the rate is proportional to the current number of fish present, with proportionality constant [latex]k[\/latex], as<\/p>\r\n

[latex]P\\prime =0.4P\\left(1-\\frac{P}{10000}\\right)-kP[\/latex].<\/p>\r\n\r\n

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17. [T]<\/strong> For the previous fishing problem, draw a directional field assuming [latex]k=0.1[\/latex]. Draw some solutions that exhibit this behavior. What are the equilibria and what are their stabilities?<\/div>\r\n<\/div>\r\n<\/div>\r\n
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18. [T]<\/strong> Use software or a calculator to draw directional fields for [latex]k=0.4[\/latex]. What are the nonnegative equilibria and their stabilities?<\/p>\r\n[reveal-answer q=\"349929\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"349929\"]\"A[latex]{P}_{1}=0[\/latex] is semi-stable[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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19. [T]<\/strong> Use software or a calculator to draw directional fields for [latex]k=0.6[\/latex]. What are the equilibria and their stabilities?<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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20.\u00a0<\/strong>Solve this equation, assuming a value of [latex]k=0.05[\/latex] and an initial condition of [latex]2000[\/latex] fish.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"995057\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"995057\"][latex]y=\\frac{-20}{4\\times {10}^{-6}-0.002{e}^{0.01t}}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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21.\u00a0<\/strong>Solve this equation, assuming a value of [latex]k=0.05[\/latex] and an initial condition of [latex]5000[\/latex] fish.<\/div>\r\n<\/div>\r\n<\/div>\r\n

The following problems add in a minimal threshold value for the species to survive, [latex]T[\/latex], which changes the differential equation to [latex]P\\prime \\left(t\\right)=rP\\left(1-\\frac{P}{K}\\right)\\left(1-\\frac{T}{P}\\right)[\/latex].<\/p>\r\n\r\n

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22.\u00a0<\/strong>Draw the directional field of the threshold logistic equation, assuming [latex]K=10,r=0.1,T=2[\/latex]. When does the population survive? When does it go extinct?<\/p>\r\n[reveal-answer q=\"693034\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"693034\"]\"A[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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23.\u00a0<\/strong>For the preceding problem, solve the logistic threshold equation, assuming the initial condition [latex]P\\left(0\\right)={P}_{0}[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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24.\u00a0<\/strong>Bengal tigers in a conservation park have a carrying capacity of [latex]100[\/latex] and need a minimum of [latex]10[\/latex] to survive. If they grow in population at a rate of [latex]1\\text{%}[\/latex] per year, with an initial population of [latex]15[\/latex] tigers, solve for the number of tigers present.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"390180\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"390180\"][latex]P\\left(t\\right)=\\frac{850+500{e}^{0.009t}}{85+5{e}^{0.009t}}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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25.\u00a0<\/strong>A forest containing ring-tailed lemurs in Madagascar has the potential to support [latex]5000[\/latex] individuals, and the lemur population grows at a rate of [latex]5\\text{%}[\/latex] per year. A minimum of [latex]500[\/latex] individuals is needed for the lemurs to survive. Given an initial population of [latex]600[\/latex] lemurs, solve for the population of lemurs.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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26. <\/strong>The population of mountain lions in Northern Arizona has an estimated carrying capacity of [latex]250[\/latex] and grows at a rate of [latex]0.25\\text{%}[\/latex] per year and there must be [latex]25[\/latex] for the population to survive. With an initial population of [latex]30[\/latex] mountain lions, how many years will it take to get the mountain lions off the endangered species list (at least [latex]100[\/latex]?)<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"655590\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"655590\"][latex]13[\/latex] years months[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

The following questions consider the Gompertz equation<\/span>, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells.<\/p>\r\n\r\n

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27.\u00a0<\/strong>The Gompertz equation is given by [latex]P\\left(t\\right)\\prime =\\alpha \\text{ln}\\left(\\frac{K}{P\\left(t\\right)}\\right)P\\left(t\\right)[\/latex]. Draw the directional fields for this equation assuming all parameters are positive, and given that [latex]K=1[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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28. <\/strong>Assume that for a population, [latex]K=1000[\/latex] and [latex]\\alpha =0.05[\/latex]. Draw the directional field associated with this differential equation and draw a few solutions. What is the behavior of the population?<\/p>\r\n[reveal-answer q=\"791416\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"791416\"]\"A[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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29.\u00a0<\/strong>Solve the Gompertz equation for generic [latex]\\alpha [\/latex] and [latex]K[\/latex] and [latex]P\\left(0\\right)={P}_{0}[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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30. [T]<\/strong> The Gompertz equation has been used to model tumor growth in the human body. Starting from one tumor cell on day [latex]1[\/latex] and assuming [latex]\\alpha =0.1[\/latex] and a carrying capacity of [latex]10[\/latex] million cells, how long does it take to reach \"detection\" stage at [latex]5[\/latex] million cells?<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"666872\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"666872\"][latex]31.465[\/latex] days[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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31. [T]<\/strong> It is estimated that the world human population reached [latex]3[\/latex] billion people in [latex]1959[\/latex] and [latex]6[\/latex] billion in [latex]1999[\/latex]. Assuming a carrying capacity of [latex]16[\/latex] billion humans, write and solve the differential equation for logistic growth, and determine what year the population reached [latex]7[\/latex] billion.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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32. [T]<\/strong> It is estimated that the world human population reached [latex]3[\/latex] billion people in [latex]1959[\/latex] and [latex]6[\/latex] billion in [latex]1999[\/latex]. Assuming a carrying capacity of [latex]16[\/latex] billion humans, write and solve the differential equation for Gompertz growth, and determine what year the population reached [latex]7[\/latex] billion. Was logistic growth or Gompertz growth more accurate, considering world population reached [latex]7[\/latex] billion on October [latex]31,2011?[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"264651\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"264651\"]September [latex]2008[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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33.\u00a0<\/strong>Show that the population grows fastest when it reaches half the carrying capacity for the logistic equation [latex]P\\prime =rP\\left(1-\\frac{P}{K}\\right)[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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34.\u00a0<\/strong>When does population increase the fastest in the threshold logistic equation [latex]P\\prime \\left(t\\right)=rP\\left(1-\\frac{P}{K}\\right)\\left(1-\\frac{T}{P}\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"552720\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"552720\"][latex]\\frac{K+T}{2}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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35.\u00a0<\/strong>When does population increase the fastest for the Gompertz equation [latex]P\\left(t\\right)\\prime =\\alpha \\text{ln}\\left(\\frac{K}{P\\left(t\\right)}\\right)P\\left(t\\right)?[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

Below is a table of the populations of whooping cranes in the wild from [latex]1940\\text{ to }2000[\/latex]. The population rebounded from near extinction after conservation efforts began. The following problems consider applying population models to fit the data. Assume a carrying capacity of [latex]10,000[\/latex] cranes. Fit the data assuming years since [latex]1940[\/latex] (so your initial population at time [latex]0[\/latex] would be [latex]22[\/latex] cranes).<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Source:<\/em> https:\/\/www.savingcranes.org\/images\/stories\/site_images\/conservation\/whooping_crane\/pdfs\/historic_wc_numbers.pdf<\/caption>\r\n
Year (years since conservation began)<\/th>\r\nWhooping Crane Population<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
[latex]1940\\left(0\\right)[\/latex]<\/td>\r\n[latex]22[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]1950\\left(10\\right)[\/latex]<\/td>\r\n[latex]31[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]1960\\left(20\\right)[\/latex]<\/td>\r\n[latex]36[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]1970\\left(30\\right)[\/latex]<\/td>\r\n[latex]57[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]1980\\left(40\\right)[\/latex]<\/td>\r\n[latex]91[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]1990\\left(50\\right)[\/latex]<\/td>\r\n[latex]159[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]2000\\left(60\\right)[\/latex]<\/td>\r\n[latex]256[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
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36.\u00a0<\/strong>Find the equation and parameter [latex]r[\/latex] that best fit the data for the logistic equation.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"903063\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"903063\"][latex]r=0.0405[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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37.\u00a0<\/strong>Find the equation and parameters [latex]r[\/latex] and [latex]T[\/latex] that best fit the data for the threshold logistic equation.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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38.\u00a0<\/strong>Find the equation and parameter [latex]\\alpha [\/latex] that best fit the data for the Gompertz equation.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"783037\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"783037\"][latex]\\alpha =0.0081[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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39.\u00a0<\/strong>Graph all three solutions and the data on the same graph. Which model appears to be most accurate?<\/div>\r\n<\/div>\r\n<\/div>\r\n
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40.\u00a0<\/strong>Using the three equations found in the previous problems, estimate the population in [latex]2010[\/latex] (year [latex]70[\/latex] after conservation). The real population measured at that time was [latex]437[\/latex]. Which model is most accurate?<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"776528\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"776528\"]Logistic: [latex]361[\/latex], Threshold: [latex]436[\/latex], Gompertz: [latex]309[\/latex].[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

For the following problems, consider the logistic equation in the form [latex]P\\prime =CP-{P}^{2}[\/latex]. Draw the directional field and find the stability of the equilibria.<\/p>\n

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1.\u00a0<\/strong>[latex]C=3[\/latex]<\/div>\n<\/div>\n<\/div>\n
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2.\u00a0<\/strong>[latex]C=0[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
\"A[latex]P=0[\/latex] semi-stable<\/div>\n<\/div>\n

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3.\u00a0<\/strong>[latex]C=-3[\/latex]<\/div>\n<\/div>\n<\/div>\n
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4.\u00a0<\/strong>Solve the logistic equation for [latex]C=10[\/latex] and an initial condition of [latex]P\\left(0\\right)=2[\/latex].<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]P=\\frac{10{e}^{10x}}{{e}^{10x}+4}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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5.\u00a0<\/strong>Solve the logistic equation for [latex]C=-10[\/latex] and an initial condition of [latex]P\\left(0\\right)=2[\/latex].<\/div>\n<\/div>\n<\/div>\n
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6.\u00a0<\/strong>A population of deer inside a park has a carrying capacity of [latex]200[\/latex] and a growth rate of [latex]2\\text{%}[\/latex]. If the initial population is [latex]50[\/latex] deer, what is the population of deer at any given time?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]P\\left(t\\right)=\\frac{10000{e}^{0.02t}}{150+50{e}^{0.02t}}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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7.\u00a0<\/strong>A population of frogs in a pond has a growth rate of [latex]5\\text{%}[\/latex]. If the initial population is [latex]1000[\/latex] frogs and the carrying capacity is [latex]6000[\/latex], what is the population of frogs at any given time?<\/div>\n<\/div>\n<\/div>\n
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8. [T]<\/strong> Bacteria grow at a rate of [latex]20\\text{%}[\/latex] per hour in a petri dish. If there is initially one bacterium and a carrying capacity of [latex]1[\/latex] million cells, how long does it take to reach [latex]500,000[\/latex] cells?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]69[\/latex] hours [latex]5[\/latex] minutes<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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9. [T]<\/strong> Rabbits in a park have an initial population of [latex]10[\/latex] and grow at a rate of [latex]4\\text{%}[\/latex] per year. If the carrying capacity is [latex]500[\/latex], at what time does the population reach [latex]100[\/latex] rabbits?<\/div>\n<\/div>\n<\/div>\n
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10. [T]<\/strong> Two monkeys are placed on an island. After [latex]5[\/latex] years, there are [latex]8[\/latex] monkeys, and the estimated carrying capacity is [latex]25[\/latex] monkeys. When does the population of monkeys reach [latex]16[\/latex] monkeys?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]8[\/latex] years [latex]11[\/latex] months<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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11. [T]<\/strong> A butterfly sanctuary is built that can hold [latex]2000[\/latex] butterflies, and [latex]400[\/latex] butterflies are initially moved in. If after [latex]2[\/latex] months there are now [latex]800[\/latex] butterflies, when does the population get to [latex]1500[\/latex] butterflies?<\/div>\n<\/div>\n<\/div>\n

The following problems consider the logistic equation with an added term for depletion, either through death or emigration.<\/p>\n

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12. [T]<\/strong> The population of trout in a pond is given by [latex]P\\prime =0.4P\\left(1-\\frac{P}{10000}\\right)-400[\/latex], where [latex]400[\/latex] trout are caught per year. Use your calculator or computer software to draw a directional field and draw a few sample solutions. What do you expect for the behavior?<\/p>\n

Show Solution<\/span><\/p>\n
\"A<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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13.\u00a0<\/strong>In the preceding problem, what are the stabilities of the equilibria [latex]0<{P}_{1}<{P}_{2}?[\/latex]<\/div>\n<\/div>\n<\/div>\n
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14. [T]<\/strong> For the preceding problem, use software to generate a directional field for the value [latex]f=400[\/latex]. What are the stabilities of the equilibria?<\/p>\n

Show Solution<\/span><\/p>\n
\"A[latex]{P}_{1}[\/latex] semi-stable<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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15. [T]<\/strong> For the preceding problems, use software to generate a directional field for the value [latex]f=600[\/latex]. What are the stabilities of the equilibria?<\/div>\n<\/div>\n<\/div>\n
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16. [T]<\/strong> For the preceding problems, consider the case where a certain number of fish are added to the pond, or [latex]f=-200[\/latex]. What are the nonnegative equilibria and their stabilities?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\"A<\/span>
\n[latex]{P}_{2}>0[\/latex] stable<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

It is more likely that the amount of fishing is governed by the current number of fish present, so instead of a constant number of fish being caught, the rate is proportional to the current number of fish present, with proportionality constant [latex]k[\/latex], as<\/p>\n

[latex]P\\prime =0.4P\\left(1-\\frac{P}{10000}\\right)-kP[\/latex].<\/p>\n

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17. [T]<\/strong> For the previous fishing problem, draw a directional field assuming [latex]k=0.1[\/latex]. Draw some solutions that exhibit this behavior. What are the equilibria and what are their stabilities?<\/div>\n<\/div>\n<\/div>\n
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18. [T]<\/strong> Use software or a calculator to draw directional fields for [latex]k=0.4[\/latex]. What are the nonnegative equilibria and their stabilities?<\/p>\n

Show Solution<\/span><\/p>\n
\"A[latex]{P}_{1}=0[\/latex] is semi-stable<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n

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19. [T]<\/strong> Use software or a calculator to draw directional fields for [latex]k=0.6[\/latex]. What are the equilibria and their stabilities?<\/span><\/div>\n<\/div>\n<\/div>\n
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20.\u00a0<\/strong>Solve this equation, assuming a value of [latex]k=0.05[\/latex] and an initial condition of [latex]2000[\/latex] fish.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=\\frac{-20}{4\\times {10}^{-6}-0.002{e}^{0.01t}}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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21.\u00a0<\/strong>Solve this equation, assuming a value of [latex]k=0.05[\/latex] and an initial condition of [latex]5000[\/latex] fish.<\/div>\n<\/div>\n<\/div>\n

The following problems add in a minimal threshold value for the species to survive, [latex]T[\/latex], which changes the differential equation to [latex]P\\prime \\left(t\\right)=rP\\left(1-\\frac{P}{K}\\right)\\left(1-\\frac{T}{P}\\right)[\/latex].<\/p>\n

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22.\u00a0<\/strong>Draw the directional field of the threshold logistic equation, assuming [latex]K=10,r=0.1,T=2[\/latex]. When does the population survive? When does it go extinct?<\/p>\n

Show Solution<\/span><\/p>\n
\"A<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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23.\u00a0<\/strong>For the preceding problem, solve the logistic threshold equation, assuming the initial condition [latex]P\\left(0\\right)={P}_{0}[\/latex].<\/div>\n<\/div>\n<\/div>\n
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24.\u00a0<\/strong>Bengal tigers in a conservation park have a carrying capacity of [latex]100[\/latex] and need a minimum of [latex]10[\/latex] to survive. If they grow in population at a rate of [latex]1\\text{%}[\/latex] per year, with an initial population of [latex]15[\/latex] tigers, solve for the number of tigers present.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]P\\left(t\\right)=\\frac{850+500{e}^{0.009t}}{85+5{e}^{0.009t}}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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25.\u00a0<\/strong>A forest containing ring-tailed lemurs in Madagascar has the potential to support [latex]5000[\/latex] individuals, and the lemur population grows at a rate of [latex]5\\text{%}[\/latex] per year. A minimum of [latex]500[\/latex] individuals is needed for the lemurs to survive. Given an initial population of [latex]600[\/latex] lemurs, solve for the population of lemurs.<\/div>\n<\/div>\n<\/div>\n
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26. <\/strong>The population of mountain lions in Northern Arizona has an estimated carrying capacity of [latex]250[\/latex] and grows at a rate of [latex]0.25\\text{%}[\/latex] per year and there must be [latex]25[\/latex] for the population to survive. With an initial population of [latex]30[\/latex] mountain lions, how many years will it take to get the mountain lions off the endangered species list (at least [latex]100[\/latex]?)<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]13[\/latex] years months<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

The following questions consider the Gompertz equation<\/span>, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells.<\/p>\n

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27.\u00a0<\/strong>The Gompertz equation is given by [latex]P\\left(t\\right)\\prime =\\alpha \\text{ln}\\left(\\frac{K}{P\\left(t\\right)}\\right)P\\left(t\\right)[\/latex]. Draw the directional fields for this equation assuming all parameters are positive, and given that [latex]K=1[\/latex].<\/div>\n<\/div>\n<\/div>\n
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28. <\/strong>Assume that for a population, [latex]K=1000[\/latex] and [latex]\\alpha =0.05[\/latex]. Draw the directional field associated with this differential equation and draw a few solutions. What is the behavior of the population?<\/p>\n

Show Solution<\/span><\/p>\n
\"A<\/div>\n<\/div>\n

<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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29.\u00a0<\/strong>Solve the Gompertz equation for generic [latex]\\alpha[\/latex] and [latex]K[\/latex] and [latex]P\\left(0\\right)={P}_{0}[\/latex].<\/div>\n<\/div>\n<\/div>\n
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30. [T]<\/strong> The Gompertz equation has been used to model tumor growth in the human body. Starting from one tumor cell on day [latex]1[\/latex] and assuming [latex]\\alpha =0.1[\/latex] and a carrying capacity of [latex]10[\/latex] million cells, how long does it take to reach “detection” stage at [latex]5[\/latex] million cells?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]31.465[\/latex] days<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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31. [T]<\/strong> It is estimated that the world human population reached [latex]3[\/latex] billion people in [latex]1959[\/latex] and [latex]6[\/latex] billion in [latex]1999[\/latex]. Assuming a carrying capacity of [latex]16[\/latex] billion humans, write and solve the differential equation for logistic growth, and determine what year the population reached [latex]7[\/latex] billion.<\/div>\n<\/div>\n<\/div>\n
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32. [T]<\/strong> It is estimated that the world human population reached [latex]3[\/latex] billion people in [latex]1959[\/latex] and [latex]6[\/latex] billion in [latex]1999[\/latex]. Assuming a carrying capacity of [latex]16[\/latex] billion humans, write and solve the differential equation for Gompertz growth, and determine what year the population reached [latex]7[\/latex] billion. Was logistic growth or Gompertz growth more accurate, considering world population reached [latex]7[\/latex] billion on October [latex]31,2011?[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
September [latex]2008[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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33.\u00a0<\/strong>Show that the population grows fastest when it reaches half the carrying capacity for the logistic equation [latex]P\\prime =rP\\left(1-\\frac{P}{K}\\right)[\/latex].<\/div>\n<\/div>\n<\/div>\n
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34.\u00a0<\/strong>When does population increase the fastest in the threshold logistic equation [latex]P\\prime \\left(t\\right)=rP\\left(1-\\frac{P}{K}\\right)\\left(1-\\frac{T}{P}\\right)?[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{K+T}{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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35.\u00a0<\/strong>When does population increase the fastest for the Gompertz equation [latex]P\\left(t\\right)\\prime =\\alpha \\text{ln}\\left(\\frac{K}{P\\left(t\\right)}\\right)P\\left(t\\right)?[\/latex]<\/div>\n<\/div>\n<\/div>\n

Below is a table of the populations of whooping cranes in the wild from [latex]1940\\text{ to }2000[\/latex]. The population rebounded from near extinction after conservation efforts began. The following problems consider applying population models to fit the data. Assume a carrying capacity of [latex]10,000[\/latex] cranes. Fit the data assuming years since [latex]1940[\/latex] (so your initial population at time [latex]0[\/latex] would be [latex]22[\/latex] cranes).<\/p>\n\n\n\n\n\n\n\n\n\n\n\n
Source:<\/em> https:\/\/www.savingcranes.org\/images\/stories\/site_images\/conservation\/whooping_crane\/pdfs\/historic_wc_numbers.pdf<\/caption>\n
Year (years since conservation began)<\/th>\nWhooping Crane Population<\/th>\n<\/tr>\n<\/thead>\n
[latex]1940\\left(0\\right)[\/latex]<\/td>\n[latex]22[\/latex]<\/td>\n<\/tr>\n
[latex]1950\\left(10\\right)[\/latex]<\/td>\n[latex]31[\/latex]<\/td>\n<\/tr>\n
[latex]1960\\left(20\\right)[\/latex]<\/td>\n[latex]36[\/latex]<\/td>\n<\/tr>\n
[latex]1970\\left(30\\right)[\/latex]<\/td>\n[latex]57[\/latex]<\/td>\n<\/tr>\n
[latex]1980\\left(40\\right)[\/latex]<\/td>\n[latex]91[\/latex]<\/td>\n<\/tr>\n
[latex]1990\\left(50\\right)[\/latex]<\/td>\n[latex]159[\/latex]<\/td>\n<\/tr>\n
[latex]2000\\left(60\\right)[\/latex]<\/td>\n[latex]256[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
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36.\u00a0<\/strong>Find the equation and parameter [latex]r[\/latex] that best fit the data for the logistic equation.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]r=0.0405[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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37.\u00a0<\/strong>Find the equation and parameters [latex]r[\/latex] and [latex]T[\/latex] that best fit the data for the threshold logistic equation.<\/div>\n<\/div>\n<\/div>\n
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38.\u00a0<\/strong>Find the equation and parameter [latex]\\alpha[\/latex] that best fit the data for the Gompertz equation.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\alpha =0.0081[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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39.\u00a0<\/strong>Graph all three solutions and the data on the same graph. Which model appears to be most accurate?<\/div>\n<\/div>\n<\/div>\n
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40.\u00a0<\/strong>Using the three equations found in the previous problems, estimate the population in [latex]2010[\/latex] (year [latex]70[\/latex] after conservation). The real population measured at that time was [latex]437[\/latex]. Which model is most accurate?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
Logistic: [latex]361[\/latex], Threshold: [latex]436[\/latex], Gompertz: [latex]309[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t
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Candela Citations<\/h3>\n\t\t\t\t\t
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