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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A population growing exponentially has a population curve of characteristic
shape!
Time
ln (
N
)
J-shaped curve