Chapter 11.7, Problem 48STP

(a)

To determine

Find the population of the wolves after one year, evaluate $p_1=0.45+1.5(0.45)(1-0.45)$

Answer to Problem 48STP

The population of wolves after 1 year is 0.82125

Explanation of Solution

Given:

In a particular forest, scientists are interested in how the population of wolves will change over the next two years. One model for animal population is the Verhulst population model, $p_{n+1}=p_n+r{p}_n(1-p_n)$ where n represents the number of time periods that have passes $p_n$ represent the percent of the maximum sustainable population that exists at time n, and r is the growth factor.

To find the population of the wolves after one year, evaluate $p_1=0.45+1.5(0.45)(1-0.45)$

Concept Used:

Evaluate: $p_1=0.45+1.5(0.45)(1-0.45)$

Calculation:

  $p_1=0.45+1.5(0.45)(1-0.45)=0.45+0.675·0.55=0.82125$

Thus, the population of wolves after 1 year is 0.82125

(b)

To determine

Explain what each number in the expression in part a represents.

Answer to Problem 48STP

0.45 represents the maximum sustainable population of 45 % and 1.5 is the growth factor.

Explanation of Solution

Given:

In a particular forest, scientists are interested in how the population of wolves will change over the next two years. One model for animal population is the Verhulst population model, $p_{n+1}=p_n+r{p}_n(1-p_n)$ where n represents the number of time periods that have passes $p_n$ represent the percent of the maximum sustainable population that exists at time n, and r is the growth factor.

Explain what each number in the expression in part a represents.

Concept Used:

0.45 represents the maximum sustainable population of 45 % and 1.5 is the growth factor.

Thus, 0.45 represents the maximum sustainable population of 45 % and 1.5 is the growth factor.

(c)

To determine

Find the new population by multiplying 165 by the value in part a.

Answer to Problem 48STP

The new population of wolves is about 136.

Explanation of Solution

Given:

In a particular forest, scientists are interested in how the population of wolves will change over the next two years. One model for animal population is the Verhulst population model, $p_{n+1}=p_n+r{p}_n(1-p_n)$ where n represents the number of time periods that have passes $p_n$ represent the percent of the maximum sustainable population that exists at time n, and r is the growth factor.

The current population of wolves is 165. Find the new population by multiplying 165 by the value in part a.

Concept Used:

New population = $165·0.82125\approx 136$ wolves.

Thus, the new population of wolves is about 136.