SME
Differential Equations · Problem type
Logistic growth
Use this for “A pond can hold 2000 fish. It is stocked with 200 and grows by dP/dt = 0.3P(1 − P/2000), t in years. How many fish are there after 5 years?”
Chart
Horizontal asymptotes P = 0 and P = 1,000
P(4) ≈ 450.9
This is an example. Type your own values to start a new problem.
dP
dt
=
r
P(1
−
P/
K
)
P(0)
=
P₀
P(
t (years)
)
=
P
A
Constant in the solution
worked out (not typed)
G
Rate of growth at time t
worked out (not typed)
Formulas
A
=
K
−
P₀
P₀
9
=
1,000 − 100
100
P
=
K
1 +
A
×
e
−
r
×
t
450.8531
=
1,000
1 + 9 ×
e
−0.5 × 4
K
×
(1
−
P
P₀
×
e
−
r
×
t
)
=
P
×
(1
−
e
−
r
×
t
)
1,000
×
(1
−
450.8531
100
×
e
−0.5 × 4
)
=
450.8531
×
(1
−
e
−0.5 × 4
)
G
=
r
×
P
×
(1
−
P
K
)
123.7923
=
0.5
×
450.8531
×
(1
−
450.8531
1,000
)
Assumptions
•
dP/dt = rP(1 − P/K): near 0 the population grows at rate r, and the growth slows to 0 as P nears the carrying capacity K.
•
Separating and splitting 1 ÷ (P(1 − P/K)) by partial fractions gives P = K ÷ (1 + Ae−rt), with A = (K − P₀) ÷ P₀ from P(0) = P₀.
•
P₀ > 0. A start above K falls toward K instead of rising, and P never crosses K.
•
r and t use the same time unit: r per year with t in years.
Step-by-step
WE KNOW
K = 1,000, P₀ = 100, r = 0.5, t = 4 years
FIND
constant in the solution (A), size at time t (P), rate of growth at time t (G)
Step 1 · Find constant in the solution (A)
Use A = (K − P₀) ÷ P₀
Put
t
=
0
into
P
=
K
1 +
A
e
−
rt
:
P₀
=
K
1 +
A
,
so
A
=
K
−
P₀
P₀
.
A
=
1,000 − 100
100
A
=
900
100
A
=
9
Step 2 · Find size at time t (P)
Use P = K ÷ (1 + A × e−r × t)
The
term
A
e
−
rt
shrinks
as
t
grows,
so
P
closes
in
on
K
.
dP/dt
=
0.5
P
(1
−
P
1,000
)
P
(
t
)
=
1,000
1 + 9
e
−0.5
t
P
(4)
=
1,000
1 + 9e⁻²
=
1,000
1 + 9 × 0.135335
=
450.8531
P
=
450.8531
Step 3 · Find rate of growth at time t (G)
Use G = r × P × (1 − P ÷ K)
Put
P
into
the
equation
itself:
the
growth
is
fastest
at
P
=
K
2
and
falls
to
0
at
K
.
G
=
0.5
×
450.8531
×
(1
−
450.8531
1,000
)
G
=
0.5
×
450.8531
×
(1
−
0.45085)
G
=
0.5
×
450.8531
×
0.54915
G
=
225.43
×
0.54915
G
=
123.7923
Check
Put the numbers back in. Both sides should match.
9
=
1,000 − 100
100
✓
450.8531
=
1,000
1 + 9 ×
e
−0.5 × 4
✓
1,000
×
(1
−
450.8531
100
×
e
−0.5 × 4
)
=
450.8531
×
(1
−
e
−0.5 × 4
)
✓
123.7923
=
0.5
×
450.8531
×
(1
−
450.8531
1,000
)
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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