Question

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

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

where t is measured in years

(a) Find the initial bird population

(b) Draw a graph of the function $n\left(t\right)$

(c) What size does the population approach as time goes on?

Short answer

a. The initial bird population is 200.

b. The required graph is:

c. The size that the population approach as time goes on is 11200

Step-by-step solution

Part a. Step 1. Given.

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

Part a. Step 2. To determine.

We have to find the initial bird population.

Part a. Step 3. Calculation.

For initial time $t=0$.

So, for$t=0$ the model becomes:

$$\begin{gathered} n\left(0\right)=\frac{5600}{0.5+27.5e^{-0.0440}} \\ =\frac{5600}{0.5+27.5e^0} \\ =\frac{5600}{0.5+27.5\left(1\right)} \\ =200 \end{gathered}$$

Hence, the initial bird population is 200.

Part b. Step 1. Given.

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

Part b. Step 2. To determine.

We have to draw a graph of the given function.

Part b. Step 3. Calculation.

We’ll enter the given expression into a graphing calculator.

Here, we are going to use Desmos calculator. In this calculator we replace t by x to get the graph.

The required graph is:

Part c. Step 1. Given.

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

Part c. Step 2. To determine.

We have to find the size that the population approach as time goes on.

Part c. Step 3. Calculation.

From the graph in part b, the horizontal asymptote is at $n\left(t\right)=11200$.

It means the size that the population approach as time goes on is 11200.