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Q3.2-15E

Expert-verifiedFound in: Page 101

Book edition
9th

Author(s)
R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages
616 pages

ISBN
9780321977069

**In Problem 14, suppose we have the additional information that the population of alligators on the grounds of the Kennedy Space Center in 1993 was estimated to be 4100. Use a logistic model to estimate the population of ****alligators in the year 2020. What is the predicted limiting population? [Hint: Use the formulas in Problem 12.**

The estimated population of alligators in the year 2020 is **6572** and the predicted limiting population is **6693.**

Given, that in 1980, the population of alligators on the Kennedy Space Center grounds was estimated to be 1500 and it was estimated to be 4100 in 1993 and 6000 in 2006. We have to find estimated population of alligators in the year 2020 and the predicting limiting population.

Here, we have initial population, ${p}_{0}=1500$

${p}_{a}=4100\phantom{\rule{0ex}{0ex}}{p}_{b}=6000$

${t}_{a}=13$ (Because, 1993-1980=13)

${t}_{b}=26$(Because, 2006-1980=26)

We will use the following formula to find the estimated population of alligators in the year 2020,

${\mathit{p}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}{\mathbf{=}}\frac{{\mathbf{p}}_{\mathbf{0}}{\mathbf{p}}_{\mathbf{1}}}{{\mathbf{p}}_{\mathbf{0}}\mathbf{+}\mathbf{(}{\mathbf{p}}_{\mathbf{1}}\mathbf{-}{\mathbf{p}}_{\mathbf{0}}\mathbf{)}{\mathbf{e}}^{\mathbf{-}{\mathbf{Ap}}_{\mathbf{1}}\mathbf{t}}}{\mathbf{\xb7}}{\mathbf{\xb7}}{\mathbf{\xb7}}{\mathbf{\xb7}}{\mathbf{\xb7}}{\mathbf{\xb7}}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$

To find the values of and A, we will use the following formulas from problem 12,

${{\mathit{p}}}_{{\mathbf{1}}}{\mathbf{=}}\mathbf{\left[}\frac{{\mathbf{p}}_{\mathbf{a}}{\mathbf{p}}_{\mathbf{b}}\mathbf{-}\mathbf{2}{\mathbf{p}}_{\mathbf{0}}{\mathbf{p}}_{\mathbf{b}}\mathbf{+}{\mathbf{p}}_{\mathbf{0}}{\mathbf{p}}_{\mathbf{a}}}{{\mathbf{p}}_{\mathbf{a}}^{\mathbf{2}}\mathbf{-}{\mathbf{p}}_{\mathbf{0}}{\mathbf{p}}_{\mathbf{b}}}\mathbf{\right]}{{\mathit{p}}}_{{\mathbf{a}}}{\mathbf{,}}{\mathbf{\xb7}}{\mathbf{\xb7}}{\mathbf{\xb7}}{\mathbf{\xb7}}{\mathbf{\xb7}}{\mathbf{\xb7}}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\phantom{\rule{0ex}{0ex}}{\mathit{A}}{\mathbf{=}}\frac{\mathbf{1}}{{\mathbf{p}}_{\mathbf{1}}{\mathbf{t}}_{\mathbf{a}}}{\mathit{l}}{\mathit{n}}\mathbf{\left[}\frac{{\mathbf{p}}_{\mathbf{b}}\left({p}_{a}-{p}_{0}\right)}{{\mathbf{p}}_{\mathbf{0}}\left({p}_{b}-{p}_{a}\right)}\mathbf{\right]}{\mathbf{\xb7}}{\mathbf{\xb7}}{\mathbf{\xb7}}{\mathbf{\xb7}}{\mathbf{\xb7}}{\mathbf{\xb7}}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\phantom{\rule{0ex}{0ex}}$

We will find the values of** p _{1}** and A, using the formulas from equation (2 and 3),

** ${{\mathit{p}}}_{{\mathbf{1}}}{\mathbf{=}}{\left[\frac{\left(4100\right)\left(6000\right)-2\left(1500\right)\left(6000\right)+\left(1500\right)\left(4100\right)}{{\left(4100\right)}^{2}-\left(1500\right)\left(6000\right)}\right]}{\left(4100\right)}\phantom{\rule{0ex}{0ex}}{{\mathit{p}}}_{{\mathbf{1}}}{\mathbf{=}}{\mathbf{6693}}{\mathbf{.}}{\mathbf{34}}\phantom{\rule{0ex}{0ex}}{\mathit{A}}{\mathbf{=}}\frac{\mathbf{1}}{\left(6693.34\right)\left(13\right)}{\mathit{l}}{\mathit{n}}{\left[\frac{\left(6000\right)\left(4100-1500\right)}{\left(1500\right)\left(6000-4100\right)}\right]}\phantom{\rule{0ex}{0ex}}{\mathit{A}}{\mathbf{=}}{\mathbf{0}}{\mathbf{.}}{\mathbf{00001954}}\phantom{\rule{0ex}{0ex}}$**

One will use these values of p_{1} and A in equation (1) to find the estimated population of splake in the year 2020.

** **

To find the estimated population of alligators in the year 2020, we will substitute t=40 and other values from step1 and step3,

** ${\mathit{p}}{\left(40\right)}{\mathbf{=}}\frac{\left(1500\right)\left(6693.34\right)}{\left(1500\right)\mathbf{+}\left(6693.34-1500\right){\mathbf{e}}^{\mathbf{-}\left(0.00001954\right)\left(6693.34\right)\left(40\right)}}\phantom{\rule{0ex}{0ex}}{\mathit{p}}{\left(40\right)}{\mathbf{=}}{\mathbf{6572}}\phantom{\rule{0ex}{0ex}}$**

** ****Hence, the estimated population of alligators in the year 2020 is 6572**.

** ****Thus, the ****predicted limiting population is 6693.**

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