A breeding group of foxes is introduced into a protected area and exhibits logistic population growth. After t years the number of foxes is given by

N(t) =

37.5

0.25 + 0.76^t

foxes.

(a) How many foxes were introduced into the protected area?
  foxes

(b) Calculate N(6). (Round your answer to the nearest whole number.)
N(6) =

Explain the meaning of the number you have calculated.
This means that after 6 years there were about  foxes in the protected area.

(c) Explain how the population varies with time. Include in your explanation the average rate of increase over the first 10-year period and the average rate of increase over the second 10-year period.

The fox population is constant.The fox population is growing, but the population does not grow as rapidly in later years as it did early on.    The fox population is growing, but the population does not grow as rapidly in early years as it did later on.

(d) Find the carrying capacity for foxes in the protected area.
foxes

(e) As we saw in the discussion of terminal velocity for a skydiver, the question of when the carrying capacity is reached may lead to an involved discussion. We ask the question differently. When is 97% of carrying capacity reached? (Round your answer to the nearest year.)
after  years

check_circle
Expert Answer

thumb_up

thumb_down

Step 1

Note:

Hi there! Thank you for posting the question. As there are multiple sub-parts, according to our policy we have solved the first three sub-parts. If you need help with other sub-parts, please re-post the questions separately.

Step 2

a)

The logistic population growth is defined as,

N(t) = 37.5/(0.25 + 0.76^t), where t is number of years and N(t) is the number of foxes at year t.

The number of foxes in protected area implies that the number of foxes at initial stage.

Substitute t = 0 in the logistic population growth.

Hence,

N(0) = 37.5/(0.25 + 0.76⁰)

         = 37.5/(0.25 + 1)

        =37.5/1.25

        = 30.

Thus, 30 foxes were introduced into the protected area.

Step 3

b)

Substitute t = 6 in the logistic population growth.

Hence,

N(6) = 37.5/(0.25 + 0.76⁶)

         = 37.5/(0.25 + 0.1927)

        =37.5/0.4427

        = 84.70

        ≈ 85.

Thus, N(6) = 85.

It implies that after 6 years there were about 85 foxes in the protected area.

Step 4

c)

Substitute t = 10 in the logistic population growth.

Hence,

N(10) = 37.5/(0.25 + 0.76¹⁰)

         = 37.5/(0.25 + 0.0643)

        =37.5/0.3143

        = 119.31

        ≈ 119.

Thus, N(10) = 119.

From Part (a) it is obtained that, N(0) = 30.

Thus, the rate of change for the first 10 year period is,

N(10) – N(0)/10 = (119 – 30)/10 = 8.9.

Substitute t = 20 in the logistic population growth.

Hence,

N(20) = 37.5/(0.25 + 0.76²⁰)

         = 37.5/(0.25 + 0.00413)

        =37.5/0.0.25413

        = 147.56

        ≈ 148.

Thus, N(20) = 148.

Thus, the rate of change for the next 10 year period is,

N(20) – N(10)/10 = (148 119 30)/10 = 2.9.

Hence, it can be said that the fox population is growing, but the population does not grow as rapidly in later years as it did early on.

*Need answers to d and e

(d) Find the carrying capacity for foxes in the protected area.
foxes

(e) As we saw in the discussion of terminal velocity for a skydiver, the question of when the carrying capacity is reached may lead to an involved discussion. We ask the question differently. When is 97% of carrying capacity reached? (Round your answer to the nearest year.)
after  years

Question

A breeding group of foxes is introduced into a protected area and exhibits logistic population growth. After t years the number of foxes is given by

N(t) =

37.5

0.25 + 0.76^t

foxes.

(a) How many foxes were introduced into the protected area?
foxes

(b) Calculate N(6). (Round your answer to the nearest whole number.)
N(6) =

Explain the meaning of the number you have calculated.

This means that after 6 years there were about foxes in the protected area.

(c) Explain how the population varies with time. Include in your explanation the average rate of increase over the first 10-year period and the average rate of increase over the second 10-year period.

The fox population is constant.The fox population is growing, but the population does not grow as rapidly in later years as it did early on. The fox population is growing, but the population does not grow as rapidly in early years as it did later on.

(d) Find the carrying capacity for foxes in the protected area.
foxes

(e) As we saw in the discussion of terminal velocity for a skydiver, the question of when the carrying capacity is reached may lead to an involved discussion. We ask the question differently. When is 97% of carrying capacity reached? (Round your answer to the nearest year.)
after years

check_circle

Expert Answer

thumb_up

thumb_down

Step 1

Note:

Hi there! Thank you for posting the question. As there are multiple sub-parts, according to our policy we have solved the first three sub-parts. If you need help with other sub-parts, please re-post the questions separately.

Step 2

a)

The logistic population growth is defined as,

N(t) = 37.5/(0.25 + 0.76^t), where t is number of years and N(t) is the number of foxes at year t.

The number of foxes in protected area implies that the number of foxes at initial stage.

Substitute t = 0 in the logistic population growth.

Hence,

N(0) = 37.5/(0.25 + 0.76⁰)

= 37.5/(0.25 + 1)

=37.5/1.25

= 30.

Thus, 30 foxes were introduced into the protected area.

Step 3

b)

Substitute t = 6 in the logistic population growth.

Hence,

N(6) = 37.5/(0.25 + 0.76⁶)

= 37.5/(0.25 + 0.1927)

=37.5/0.4427

= 84.70

≈ 85.

Thus, N(6) = 85.

It implies that after 6 years there were about 85 foxes in the protected area.

Step 4

c)

Substitute t = 10 in the logistic population growth.

Hence,

N(10) = 37.5/(0.25 + 0.76¹⁰)

= 37.5/(0.25 + 0.0643)

=37.5/0.3143

= 119.31

≈ 119.

Thus, N(10) = 119.

From Part (a) it is obtained that, N(0) = 30.

Thus, the rate of change for the first 10 year period is,

N(10) – N(0)/10 = (119 – 30)/10 = 8.9.

Substitute t = 20 in the logistic population growth.

Hence,

N(20) = 37.5/(0.25 + 0.76²⁰)

= 37.5/(0.25 + 0.00413)

=37.5/0.0.25413

= 147.56

≈ 148.

Thus, N(20) = 148.

Thus, the rate of change for the next 10 year period is,

N(20) – N(10)/10 = (148 119 30)/10 = 2.9.

Hence, it can be said that the fox population is growing, but the population does not grow as rapidly in later years as it did early on.

*Need answers to d and e

(d) Find the carrying capacity for foxes in the protected area.
foxes

(e) As we saw in the discussion of terminal velocity for a skydiver, the question of when the carrying capacity is reached may lead to an involved discussion. We ask the question differently. When is 97% of carrying capacity reached? (Round your answer to the nearest year.)
after years

Accepted Answer

Answered: Image /qna-images/answer/89ac7903-1358-4fcb-afd8-462effc05c3a.jpg