1. Logistic and Exponential Growth
Chapter 40
Logistic and Exponential Growth

2. You Must Know
The differences between exponential and logistic models of population growth.
How to calculate and graph growth

3. Per Capita Rate of Increase−
Change in
population
size
Births
Immigrants
Deaths
Emigrants  
If immigration and emigration are ignored, a population’s growth rate (per capita increase) equals birth rate minus death rate. 3

4. Population Changes number of births over a period of time Per-capita
birth rate (b) = ÷
population
1,500 ÷
10,000 =
0.15
number of deaths over a period of time
Per-capita
death rate (d) = ÷
population
500 ÷
10,000 =
0.05
Per-capita rate
of increase (r)
(b) - =
(d)
0.15 -
0.05 =
0.10 or 10%

5. r  b − d r equals 0 (population stays the same size) r is positive
(population increases)
r is negative
(population decreases) 5

6. N is the change in population size, t is the time interval.
 rN
N is the change in population size, t is the time interval.
Example
0.04 individuals per year
Population’s growth rate
N = population size =
2,000 individuals X
= an increase of 80 individuals per year 6

7. Exponential population growth is population increase under idealized conditions. Under these conditions, the rate of increase is at its maximum, denoted as rmax.

8. Exponential population growth results in a J-shaped curve.
Exponential Growth dt dN
 rmax N
Population size (N)
Number of generations
Exponential population growth results in a J-shaped curve. 8

9. 2,000 dt dN  1.0N 1,500 dt dN  0.5N Population size (N) 1,000 500 5
Figure 40.17
2,000 dt dN
 1.0N
1,500 dt dN
 0.5N
Population size (N)
1,000
500 5 10 15
Number of generations 9

10. The J-shaped curve of exponential growth characterizes populations in new environments or rebounding populations.
8,000
6,000
Elephant population
4,000
Figure Exponential growth in the African elephant population of Kruger National Park, South Africa
The elephant population in Kruger National Park, South Africa, grew exponentially after hunting was banned.
2,000
1900
1910
1920
1930
1940
1950
1960
1970
Year 10

11. Bozeman Science video on Exponential Growth

12. xkcd
This is a joke…

13. The Logistic Growth Model
Carrying capacity (K) is the maximum population size the environment can support.
K  carrying capacity
Population size (N) dt dN
(K − N) K
 rmax N
Exponential growth cannot be sustained for long in any population.
A more realistic population model limits growth by incorporating carrying capacity.
Carrying capacity varies with the abundance of limiting resources. In the logistic population growth model, the per capita rate of increase declines as carrying capacity is reached.
The logistic model starts with the exponential model and adds an expression that reduces per capita rate of increase as N approaches K.
The logistic model of population growth produces a sigmoid (S-shaped) curve.
Number of generations 13

14. Table 40.2 25
1.0
0.98
0.98
+ 25
250
1.0
0.83
0.83
+ 208
750
1.0
0.50
0.50
+ 375
1,000
1.0
0.33
0.33
+ 333
1,500
1.0
0.00
0.00 14

15. Exponential growth 2,000 dt dN  1.0N 1,500 K  1,500 Logistic growth
Figure 40.19
Exponential
growth
2,000 dt dN
 1.0N
1,500
K  1,500
Logistic growth dt dN
(1,500  N)
1,500
Population size (N)
 1.0N
1,000
Population growth
begins slowing here.
Figure Population growth predicted by the logistic model
500 5 10 15
Number of generations 15

16. Number of Paramecium/mL
Figure 40.20
1,000
180
150
800
120
Number of Daphnia/50 mL
Number of Paramecium/mL
600 90
400 60
200 30 5 10 15 20 40 60 80
100
120
140
160
Time (days)
Time (days)
The growth of many laboratory populations, including paramecia, fits an S-shaped curve when resources are limited . These organisms are grown in a constant environment lacking predators and competitors.
Some populations overshoot K before settling down to a relatively stable density.
Some populations fluctuate greatly and make it difficult to define K.
(a) A Paramecium population
in the lab
(b) A Daphnia (water flea) population in the lab 16

17. Bozeman Science video on Logistic Growth

18. You will be given this information for the test and the AP Exam