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Exponential Growth
:
Consider an organism with continuous birth and deaths (i.e., bacteria in the
gut):
generations overlap
Instead of measuring population change on a yearly or generation time scale,
we typically measure changes on very small time scales: instantaneous rates
of change.
As t
⇒
0,
Δ
N/
Δ
t
⇒
dN/dt = (b  d)N = rN
r =
instantaneous growth rate
or
intrinsic rate of increase
(individuals per
individual per unit time)
We can integrate dN/dt from time = 0 to time = t:
N
t
= N
0
e
rt
lnN
t
= lnN
0
+rt
lnN
t
lnN
0
= r
t
R
0
= e
rT
; r = ln(R
0
)/T
per generation growth rate
r < 0
, population is decreasing
r = 0,
population is stable
r > 0
, population is increasing
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View Full Document A population growing exponentially has a population curve of characteristic
shape!
Time
ln (
N
)
Jshaped curve
Population size (
N
)
d
N
dt
Population size (
N
)
d
N
1
dt
N
r
Populations growing exponentially also exhibit the following characteristics:
1)
positive feedback
of population size on population growth: larger
populations grow faster than do smaller populations
2) per capita population growth is
constant
population growth rate
vs. population size
per capita population
growth rate
vs. population size
Remember: d
N
/d
t
=
rN
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N
t
= N
0
e
rt
2N
0
= N
0
e
rt
ln
(2
N
0
/N
0
) = rt
ln(2) = rt
t
d
=
ln(2)/r
~ 0.69/r
Doubling time:
bacteria = 20 min
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This note was uploaded on 05/23/2008 for the course OCHEM 112b taught by Professor Shoeller during the Spring '08 term at UC Riverside.
 Spring '08
 shoeller

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