Chapter 6.5, Problem 24E

(a)

To determine

To find : The carrying capacity of the population.

Answer to Problem 24E

The answer is:

The carrying capacity of $700$ individuals

Explanation of Solution

Given information:

The differential equation is: $\frac{dP}{dt}=0.0008P(700-P)$

Calculation:

The logistic differential equation is:

  $\frac{dP}{dt}=kP(M-P)$

Where $M$ is maximal carrying capacity.

  $\begin{array}{c}\frac{dP}{dt}=0.0008P(700-P)\\ \Rightarrow M=700\end{array}$

Therefore the required carrying capacity of $700$ individuals.

(b)

To determine

To find: The size of the population when it is growing fastest.

Answer to Problem 24E

The answer is:

  $350$ individuals

Explanation of Solution

Given information:

The differential equation is: $\frac{dP}{dt}=0.0008P(700-P)$

Calculation:

When the population is half its carrying capacity, or $350$ people, it grows the fastest.

Therefore the required $350$ individuals of the population when it is growing fastest .

(c)

To determine

To find: The rate at that the population when it is growing fastest.

Answer to Problem 24E

The answer is:

  $98$ people per year.

Explanation of Solution

Given information:

The differential equation is: $\frac{dP}{dt}=0.0008P(700-P)$

Calculation:

The logistic differential equation is:

  $\frac{dP}{dt}=kP(M-P)$ Where $M$ is maximal carrying capacity.

Put $P=350$ into differential equation then gives:

  $\begin{array}{c}\frac{dP}{dt}=0.0008\cdot 350(700-350)\\ =0.28\cdot 350\\ =98\end{array}$

Therefore the required rate at that the population when it is growing fastest is $98$ people per year.