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Q3.2-13E

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Found in: Page 101

### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# In Problem 9, suppose we have the additional information that the population of splake in 2004 was estimated to be 5000. Use a logistic model to estimate the population of splake in the year 2020. What is the predicted limiting population? [Hint: Use the formulas in Problem 12.

The estimated population of splake in the year 2020 is 5970 and the predicted limiting population is 6000.

See the step by step solution

## Step 1: Analyzing the given statement

Given, that in 1990, the population of splake in the lake was 1000 and it was estimated to be 3000 in 1997 and 5000 in 2004. We have to find estimated population of splake in the year 2020 and the predicted limiting population.

Here, we have initial population,${p}_{0}=1000$

${p}_{a}=3000\phantom{\rule{0ex}{0ex}}{p}_{b}=5000$

${t}_{a}=7$(Because, 1997-1990=7)

${t}_{b}=14$(Because, 2004-1990=14)

## Step 2: Formulas used to find the solution

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

${\mathbit{p}}\left(t\right){\mathbf{=}}\frac{{\mathbf{p}}_{\mathbf{0}}{\mathbf{p}}_{\mathbf{1}}}{{\mathbf{p}}_{\mathbf{0}}\mathbf{+}\left({p}_{1}-{p}_{0}\right){\mathbf{e}}^{\mathbf{-}\mathbf{A}{\mathbf{p}}_{\mathbf{1}}\mathbf{t}}}{\mathbf{·}}{\mathbf{·}}{\mathbf{·}}{\mathbf{·}}{\mathbf{·}}{\mathbf{·}}\left(1\right)$

To find the values of p1 and A, we will use the following formulas from problem 12,

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

## Step 3: Determine the values of p1 and A

One will find the values of and A, using the formulas from equation (2 and 3),

${{\mathbit{p}}}_{{\mathbf{1}}}{\mathbf{=}}\left[\frac{\left(3000\right)\left(5000\right)-2\left(1000\right)\left(5000\right)+\left(1000\right)\left(3000\right)}{{\left(3000\right)}^{2}-\left(1000\right)\left(5000\right)}\right]\left(3000\right)\phantom{\rule{0ex}{0ex}}{{\mathbit{p}}}_{{\mathbf{1}}}{\mathbf{=}}{\mathbf{6000}}\phantom{\rule{0ex}{0ex}}{\mathbit{A}}{\mathbf{=}}\frac{\mathbf{1}}{\left(6000\right)\left(7\right)}{\mathbit{l}}{\mathbit{n}}\left[\frac{\left(5000\right)\left(3000-1000\right)}{\left(1000\right)\left(5000-3000\right)}\right]\phantom{\rule{0ex}{0ex}}{\mathbit{A}}{\mathbf{=}}{\mathbf{0}}{\mathbf{.}}{\mathbf{00003832}}\phantom{\rule{0ex}{0ex}}$

We will use these values of p1 and A in equation (1) to find the estimated population of splake in the year 2020.

## Step 4: Find the estimated population of splake in the year 2020

To find the estimated population of splake in the year 2020, we will substitute t=30 and other values from step1 and step3,

${\mathbit{p}}\left(30\right){\mathbf{=}}\frac{\left(1000\right)\left(6000\right)}{\left(1000\right)\mathbf{+}\left(6000-1000\right){\mathbf{e}}^{\mathbf{-}\left(0.00003832\right)\left(6000\right)\left(30\right)}}\phantom{\rule{0ex}{0ex}}{\mathbit{p}}\left(30\right){\mathbf{=}}{\mathbf{5970}}\phantom{\rule{0ex}{0ex}}$

Hence, the estimated population of splake in the year 2020 is 5970.

Thus, the predicted limiting population is 6000.

## Most popular questions for Math Textbooks

Show that ${{\mathbit{C}}}_{{\mathbf{1}}}{\mathbit{c}}{\mathbit{o}}{\mathbit{s}}{\mathbit{\omega }}{\mathbit{t}}{\mathbf{+}}{{\mathbit{C}}}_{{\mathbf{2}}}{\mathbit{s}}{\mathbit{i}}{\mathbit{n}}{\mathbit{\omega }}{\mathbit{t}}$ can be written in the form ${\mathbit{A}}{\mathbit{c}}{\mathbit{o}}{\mathbit{s}}\left(\omega t-\varphi \right)$, where ${\mathbit{A}}{\mathbf{=}}\sqrt{{\mathbf{C}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}{\mathbf{C}}_{\mathbf{2}}^{\mathbf{2}}}$ and ${\mathbit{t}}{\mathbit{a}}{\mathbit{n}}{\mathbit{\varphi }}{\mathbf{=}}{{\mathbit{C}}}_{{\mathbf{2}}}{\mathbf{/}}{{\mathbit{C}}}_{{\mathbf{1}}}$. [Hint: Use a standard trigonometric identity with ${{\mathbit{C}}}_{{\mathbf{1}}}{\mathbf{=}}{\mathbit{A}}{\mathbit{c}}{\mathbit{o}}{\mathbit{s}}{\mathbit{\varphi }}{\mathbf{,}}{{\mathbit{C}}}_{{\mathbf{2}}}{\mathbf{=}}{\mathbit{A}}{\mathbit{s}}{\mathbit{i}}{\mathbit{n}}{\mathbit{\varphi }}$.] Use this fact to verify the alternate representation (8) of F(t) discussed in Example 2 on page 104.