Q3.2-15E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

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.

Step-by-Step Solution

Answer

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

Step 1: Analyzing the given statement

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,  $p_0=1500$

 $$\begin{gathered} p_a=4100 \\ {p}_b=6000 \end{gathered}$$

 $t_a=13$ (Because, 1993-1980=13)

  $t_b=26$(Because, 2006-1980=26)

Step 2: Formulas used to find the solution

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

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

 

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

$$\begin{gathered} {\mathit{p}}_{\mathbf{1}}\mathbf{=}\mathbf{\left[}\frac{{\mathbf{p}}_{\mathbf{a}}{\mathbf{p}}_{\mathbf{b}}\mathbf{-}\mathbf{2}{\mathbf{p}}_{\mathbf{0}}{\mathbf{p}}_{\mathbf{b}}\mathbf{+}{\mathbf{p}}_{\mathbf{0}}{\mathbf{p}}_{\mathbf{a}}}{{\mathbf{p}}_{\mathbf{a}}^{\mathbf{2}}\mathbf{-}{\mathbf{p}}_{\mathbf{0}}{\mathbf{p}}_{\mathbf{b}}}\mathbf{\right]}{\mathit{p}}_{\mathbf{a}}\mathbf{,}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{\left(}\mathbf{2}\mathbf{\right)} \\ \mathit{A}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{p}}_{\mathbf{1}}{\mathbf{t}}_{\mathbf{a}}}\mathit{l}\mathit{n}\mathbf{\left[}\frac{{\mathbf{p}}_{\mathbf{b}}\left(p_a-p_0\right)}{{\mathbf{p}}_{\mathbf{0}}\left(p_b-p_a\right)}\mathbf{\right]}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{\left(}\mathbf{3}\mathbf{\right)} \end{gathered}$$

Step 3: Determine the values of p 1 and A

We will find the values of p₁  and A, using the formulas from equation (2 and 3),

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

One 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 alligators in the year 2020, we will substitute t=40 and other values from step1 and step3,

 $$\begin{gathered} \mathit{p}\left(40\right)\mathbf{=}\frac{\left(1500\right)\left(6693.34\right)}{\left(1500\right)\mathbf{+}\left(6693.34-1500\right){\mathbf{e}}^{\mathbf{-}\left(0.00001954\right)\left(6693.34\right)\left(40\right)}} \\ \mathit{p}\left(40\right)\mathbf{=}\mathbf{6572} \end{gathered}$$

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

 Thus, the predicted limiting population is 6693.