Chapter 7.5, Problem 2E

a.

To determine

To write : the logistic differential equation for the data

Answer to Problem 2E

The logistic equation is $\frac{dP}{dt}=0.005P\left(1-\frac{P}{6000}\right)$

Explanation of Solution

Given information : The data is the carrying capacity is $6000$ and $k=0.0015$ per year

Calculation :

The logistic equation is $\frac{dP}{dt}=kP\left(1-\frac{P}{M}\right)$

Where $P$ is the population

  $k$ Is the growth constant

  $M$ is carrying capacity

  $\begin{array}{l}M=6000\&k=0.0015\\ \frac{dP}{dt}=0.005P\left(1-\frac{P}{6000}\right)\end{array}$

b.

To determine

To draw : the direction field

Explanation of Solution

Given information : The data is the carrying capacity is $6000$ and $k=0.0015$ per year

Graph : The solution curves will converge to carrying capacity $M=6000$ and $P=0$ is also an equilibrium solution

The direction field is shown below

c.

To determine

To sketch : the solution of curve for initial solution and also find their significance of the inflection point

Answer to Problem 2E

Inflection occurs at appoint if the graph is continuous and the concavity of the graph reverses at that point

Explanation of Solution

Given information : The initial points are $1000,2000,4000\&8000$

Calculation :

When the initial population is $1000$ then the curve is concave down which is shown below

When the initial population is $2000$ then the curve is concave down which is shown below

When the initial population is $4000$ then the curve is concave down which is shown below

When the initial population is $8000$ then the curve is concave up which is shown below

d.

To determine

To calculate : the population after $50$ years

Answer to Problem 2E

The population after $50$ years is $1064$

Explanation of Solution

Given information : the question is

By using Euler’s method with step size $h=1$ and the initial population is $1000$

Calculation : step size $h=1$

Initial population $=1000$

To find $P\left(50\right)$

In the expression box write $0.0015\times y\times \left(1-\raisebox{1ex}{$y$}\!\left/ \!\raisebox{-1ex}{$6000$}\right.\right)$

X start is $0$

X end is $50$

Y start is $1000$

Step size $1$

So, $P\left(50\right)\approx 1064.039$

e.

To determine

To calculate : the population after $50$ years and compare with part (d)

Answer to Problem 2E

The population after $50$ years is $1064$

Explanation of Solution

Given information : The question is

If the initial population is $1000$

Calculation : when $t=50$

  $\begin{array}{l}P\left(50\right)=\frac{6000}{1+5e^{-0.0015t}}\\ P\left(50\right)=\frac{6000}{1+5e^{-0.0015\left(50\right)}}\\ P\left(50\right)=1064\end{array}$

f.

To determine

To graph : the solution of part (e)

Explanation of Solution

Given information : The question is

If the initial population is $1000$

Graph : from part (e)

  $P\left(t\right)=\frac{6000}{1+5e^{-0.0015t}}$

When the initial population is $1000$ then the graph is shown below