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 ofalligators in the year 2020. What is the predicted limiting population? [Hint: Use the formulas in Problem 12.

Short answer

The estimated population of alligators in the year 2020 is 6572 and the predicted limiting population is6693.

Step-by-step solution

Analyzing the given statement

Given, that in1980, thepopulation of alligatorson the Kennedy Space Centergroundswas estimatedto be1500 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)

Formulas used to find the solution

We will use the following formula to find theestimated 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,

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

Determine the values of  p1 and A

We will find the values ofp₁ 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.

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, thepredicted limiting population is 6693.