Chapter 14 Population Ecology • MHR 475population starts with only one individual, 0.5 h
later there will be two bacteria. Think of these as
being the offspring of a single parent, which “died”
upon reproducing. Populations like this, in which
parents and young are never present in the
population at the same time, are described as
having non-overlapping generations. Many species
of plants and insects in which individuals live for
a year or less, producing seeds or eggs just before
they die, have non-overlapping generations. In
these populations, where the next generation
actually replacesthe current one, the growth rate is
commonly referred to as the replacement rateand
symbolized as R. In the case of the bacterial
population described above, R= 2 — each individual
is replaced by two others in the next generation.
Using the replacement rate, it is possible to
develop a simple formula to predict Nat some time
in the future. The size of the population after one
generation, or N 1 (“Nat time one”) can be
calculated by multiplying the starting population
size (the number of parents) times the replacement
rate (the number of offspring each parent produced).
In mathematical terms, this would be written:
N 1 =N 0 ×R or N 1 =N 0 R
The size of the population after a second
generation, N 2 , will be N 1 (the number of individuals
available to reproduce at time 1) multiplied by the
replacement rate. That is, N 2 =N 1 R. Since we
know that N 1 =N 0 R, we can substitute to get
N 2 =(N 0 R)R=N 0 R^2. Similarly, N 3 =N 2 R, which
also equals N 0 R^3. Notice the pattern that
develops, as shown in Figure 14.11 for the first
few generations. Thus, for populations with non-
overlapping generations, the size of the population
after any number of generations (or years, or
whatever time unit is used) can be calculated
using the general formula:
Nt=N 0 Rt
This equation has been used to calculate the
size of the bacterial population existing after 20
generations in Figure 14.11. The J-shaped curve
shows how quickly this population increased.
In many species (including humans), parents
do not normally die after reproducing, and two or
more generations of organisms may be alive and
reproducing at the same time. These are called
overlapping generations. In these populations, the
per capita growth rate is the familiar ryou have
already seen. A complete explanation of the
underlying mathematics is beyond the scope of this
book. However, the formula used to predict future
population size for populations with overlapping
generations is actually quite simple to use. If N 0Time (hours) Number of bacteria Growth curve
10.0
9.5
9.0
8.5
8.0
7.5
7.0
6.5
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.01 048 576
524 288
262 144
131 072
65 536
32 768
16 384
8192
4096
2048
1024
512
256
128
64
32
16
8
4
2
1
N 5 =N 4 R=(N 0 R^4 )R=N 0 R^5
N 4 =N 3 R=(N 0 R^3 )R=N 0 R^4
N 3 =N 2 R=(N 0 R^2 )R=N 0 R^3
N 2 =N 1 R=(N 0 R)R=N 0 R^2
N 1 =N 0 R
N 0
(^0135)
1
2
3
2
4
5
6
7
8
9
10
4 6 7 8 9 10
Number of bacteria (× 100 000)
Time (hours)Figure 14.11With a replacement rate of R=2 and a generation time of 30 min,
a single bacterium would give rise to a population of over one million bacteria in
only 10 h.