SME
Ecology
Population dynamics
Use this for “With r = 0.5 a year and K = 1000, how fast does the population grow at N = 250, and when is growth fastest?”
Chart
Fastest growth at t ≈ 4.394, where it reaches half the limit, 500
Horizontal asymptotes N = 0 and N = 1,000
N(2.197) = 250
This is an example. Type your own values to start a new problem.
dN/dt (individuals/yr)
=
r (yr⁻¹)
×
N
×
−
K
N₀
Starting size
example
t
Time
calculated · yr
rK/4
Fastest growth rate
worked out (not typed) · individuals/yr
Formulas
N
=
K
÷
(1
+
(
K
−
N₀
N₀
)
×
e
−
rt
)
250
=
1,000
÷
(1
+
(
1,000 − 100
100
)
×
e
−0.5 × 2.197
)
dN/dt
=
r
×
N
×
K
−
N
K
93.75
=
0.5
×
250
×
1,000 − 250
1,000
rK
/4
=
r
×
K
4
125
=
0.5
×
1,000
4
Assumptions
•
The population grows as dN/dt = rN(K − N) ÷ K: births and deaths only, no migration.
•
r is the intrinsic rate of increase, the per-capita rate while N is small; K is the carrying capacity.
•
Growth is fastest at N = K ÷ 2, where dN/dt = rK ÷ 4; N approaches K but never passes it.
Step-by-step
WE KNOW
N = 250, r = 0.5 yr⁻¹, K = 1,000, N₀ = 100
FIND
time (t), growth rate (dN/dt), fastest growth rate (rK/4)
Step 1 · Find time (t)
Use N = K ÷ (1 + ((K − N₀) ÷ N₀) × e⁻ʳᵗ)
Solve
1
+
A
e
−
rt
=
K
N
for
e
−
rt
,
then
take
the
natural
log
and
divide
by
−
r
.
t
=
ln((
K
−
N₀
N₀
)
×
N
K
−
N
)
÷
r
t
=
ln((
1,000 − 100
100
)
×
250
1,000 − 250
)
÷
0.5
A
=
1,000 − 100
100
=
9
e
rt
=
A
×
N
K
−
N
=
9
×
250
750
=
3
t
=
ln 3
0.5
t
=
2.197
yr
Step 2 · Find growth rate (dN/dt)
Use dN/dt = r × N × (K − N) ÷ K
The
exponential
rate
rN,
slowed
by
the
share
of
K
still
unused,
K
−
N
K
.
dN/dt
=
0.5
×
250
×
1,000 − 250
1,000
dN/dt
=
0.5
×
250
×
750
1,000
dN/dt
=
125
×
750
1,000
dN/dt
=
93,750
1,000
dN/dt = 93.75 individuals/yr
Step 3 · Find fastest growth rate (rK/4)
Use rK/4 = r × K ÷ 4
Put
N
=
K
2
into
rN
(
K
−
N
)
K
:
r
×
(
K
2
)
×
K
÷ 2
K
=
rK
4
.
rK
/4
=
0.5
×
1,000
4
rK
/4
=
500
4
rK
/4
=
125
individuals/yr
(at
N
=
500,
half
of
K
)
Check
Put the numbers back in. Both sides should match.
250
=
1,000
÷
(1
+
(
1,000 − 100
100
)
×
e
−0.5 × 2.197
)
✓
93.75
=
0.5
×
250
×
1,000 − 250
1,000
✓
125
=
0.5
×
1,000
4
✓
Topic check
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