Lecture 18
Derivatives and rates of change
Rates of change in the natural sciences
Population models
We now consider the role of “rate of change” in modelling population growth. In what follows, we let
n
=
f
(
t
)
(1)
denote the number of individuals in an animal or plant population at time
t
. The change in population
between times
t
1
and
t
2
> t
1
is given by
Δ
n
=
f
(
t
2
)
−
f
(
t
1
)
.
(2)
The
average rate of change
of the population over the time interval [
t
1
, t
2
] is defined as
Δ
n
Δ
t
=
f
(
t
2
)
−
f
(
t
1
)
t
2
−
t
1
.
(3)
If we assume that the population is growing, i.e.,
f
(
t
2
)
> f
(
t
1
) for all
t
1
< t
2
, then the phrase “average
rate of change” may be replaced by “average rate of growth”. That being said, we could ignore whether
the population is growing or decaying and simply call the ratio in (3) the “average rate of growth,”
with the understanding that negative growth implies decay.
We now proceed as we have done before, letting
t
2
→
t
1
, to yield the
instantaneous rate of
growth
at
t
=
t
1
,
lim
Δ
t
→
0
Δ
n
Δ
t
=
dn
dt
=
f
′
(
t
1
)
.
(4)
There is one point that should be addressed here.
In the above discussion, we have simply
proceeded as in the past, where the dependent variable
n
is treated as a continuous real number.
Obviously, when we talk about populations, we are using integers to count the number of individuals
– there are no “halforganisms”. When the changes Δ
n
to the population for small time intervals Δ
t
are very small in comparison to the population itself, i.e., when the ratio
Δ
n
n
(5)
is very small, often written as follows,
Δ
n
n
<<
1
,
(6)
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then these changes may be viewed as continuous.
For example, when one individual is added to a
family of four, the percentage change in the number of people in the family is large, namely, a 25%
increase. When one individual is added to a city of 100,000, the percentage change is very tiny and
may be considered as infinitesimal. In the discussion that follows, we assume that sufficiently large
populations are being modelled so that
n
may be considered as a continuous variable.
Example: A culture of bacteria in a Petri dish
Suppose that we have a culture of bacterial in a Petri dish, with sufficient nutrient in the dish for the
bacteria to multiply over a number of generations. Experimentally, in such cases of “infinite resources,”
it is observed that the population of the bacteria doubles after a roughly fixed time interval
T
. Let
us assume that at time
t
= 0, there is a population of
n
(0) =
n
0
(7)
bacteria in the dish. For simplicity, let’s measure time in units of
T
– this is essentially a rescaling of
time. This means that the populations at future times
t
=
k
,
k
= 0
,
1
,
2
,
· · ·
, are given by
n
(0)
=
n
0
n
(1)
=
2
n
0
n
(2)
=
2
f
(1) = 2(2
n
0
) = 2
2
n
0
.
.
.
n
(
k
)
=
2
k
n
0
.
(8)
It is a reasonable assumption  and experimentally verifiable – that the bacteria are not doubling their
population at specific times, for example, just before the times that the experimenter measures it.
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 Fall '08
 SPEZIALE
 Calculus, Exponential Function, Derivative, Linear Approximation, Rate Of Change, Radioactive Decay, Exponential decay

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