Example 2:
Kentwood, Michigan had a population of 49,000 in the year 2013.
The infrastructure of the city
allows for a carrying capacity of 60,000 people.
r
max
= .9 for Kentwood.
a. Is the current population above or below the carrying capacity?
Will the population increase
or decrease in the next year?
Current population is below the carrying capacity so we would expect the population to
increase.
b. What will be the
population growth rate
for 2013 (include units)?
Ο
Π
Ξ
Μ
Ν
Λ
−
=
K
N
K
N
r
dt
dN
max
=
=
Ο
Π
Ξ
Μ
Ν
Λ
−
000
,
60
000
,
49
000
,
60
)
000
,
49
)(
9
.
0
(
8085 people for the year
c. What will be the
population size
at the start of 2014.
49,000 + 8085 = 57,085 people
d. Fill in the following table:
Year
Population size
Population growth rate
2013
49,000
8085
2014
57,085
2496.041625
2015
59,581
374.4296457
2016
59,955
40.04611425
2017
59,996
4.034052134
e. What happened to the population size over the years?
What happened to the population
growth rate over the years?
Population size got closer and closer to the carrying capacity of 60,000.
Population growth rate got smaller and smaller.
f. Explain your answer from part (e) using what you know about carrying capacity.
As the population gets closer and closer to the carrying capacity, the resources of the
environment start to deplete and the growth rate slows. Limiting factors take over
(especially density-dependent ones, space, food, water, shelter, etc)
g. Explain your answer from part (e) using the formula:
Ο
Π
Ξ
Μ
Ν
Λ
−
=
K
N
K
N
r
dt
dN
max
Look at this part of the formula:
Ο
Π
Ξ
Μ
Ν
Λ
−
K
N
K
As the size of the population (N) gets closer
and closer to the carrying capacity (K), then this part of the formula gets smaller and
smaller, thus reducing population growth rate.