Chapter 4.2, Problem 26E

Bird Population The population of a certain species of bird is limited by the type of habitat required for nesting. The population behaves according to the logistic growth model

$n(t)=\frac{5600}{0.5+27.5e^{-0.044t}}$

where r is measured in years.

1. (a) Find the initial bird population.
2. (b) Draw a graph of the function n(t).
3. (c) What size does the population approach as time goes on?

To determine

To find: The initial population of bird if the bird’s population follows the logistic growth model is, $n\left(t\right)=\frac{5600}{0.5+27.5e^{-0.044t}}$.

Answer to Problem 26E

The initial population of bird is 200.

Explanation of Solution

Given that the logistic growth model is $n\left(t\right)=\frac{5600}{0.5+27.5e^{-0.044t}}$.

Substitute 0 for t in $n\left(t\right)$ and calculate the initial population of bird as follows.

$\begin{array}{c}n\left(0\right)=\frac{5600}{0.5+27.5e^{-0.044\times 0}}\\ =\frac{5600}{0.5+27.5\left(1\right)}\\ =\frac{5600}{28}\\ =200\end{array}$

Hence, the initial population of bird is 200.

To determine

To sketch: The graph of the function $n\left(t\right)$.

Explanation of Solution

Given that the logistic growth model is $n\left(t\right)=\frac{5600}{0.5+27.5e^{-0.044t}}$.

Use online graphing calculator and draw the graph of the functions $n\left(t\right)=\frac{5600}{0.5+27.5e^{-0.044t}}$ as shown below in Figure 1.

From Figure 1, it is observed that the graph increasing fast and after certain point the growth is 11,200.

To determine

To evaluate: The population of bird for larger value of t.

Answer to Problem 26E

The population of bird for larger value of t is $11,200$.

Explanation of Solution

The bird population follows the logistic growth model is $n\left(t\right)=\frac{5600}{0.5+27.5e^{-0.044t}}$.

Substitute $\infty$ for t in the above formula to calculate the population of bird for larger value of t,

$\begin{array}{c}n\left(\infty \right)=\frac{5600}{0.5+27.5e^{-0.044\times \infty }}\\ =\frac{5600}{0.5+27.5{\left(e\right)}^{-\infty }}\\ =\frac{5600}{0.5}\\ =11,200\end{array}$

Hence, the population of bird for larger value of t is $11,200$.