SME
Ecology · Problem type
Life tables: R₀ and generation time
Use this for “From this life table, what are R₀ and the generation time?”
Table
Age x
lₓ
mₓ
lₓmₓ
xlₓmₓ
0
1
0
0
0
1
0.5
1
0.5
0.5
2
0.25
2
0.5
1
3
0.1
2
0.2
0.6
Σ
1.2
2.1
Generation time T = 1.75 yr
This is an example. Type your own values to start a new problem.
l₀
Survival to age 0
example
l₁
Survival to age 1
example
l₂
Survival to age 2
example
l₃
Survival to age 3
example
m₀
Fecundity at age 0
example
m₁
Fecundity at age 1
example
m₂
Fecundity at age 2
example
m₃
Fecundity at age 3
example
R₀
Net reproductive rate
worked out (not typed)
T
Generation time
worked out (not typed) · yr
Formulas
R₀
=
l₀
×
m₀
+
l₁
×
m₁
+
l₂
×
m₂
+
l₃
×
m₃
1.2 = 1 × 0 + 0.5 × 1 + 0.25 × 2 + 0.1 × 2
T
=
(0
×
l₀
×
m₀
+
1
×
l₁
×
m₁
+
2
×
l₂
×
m₂
+
3
×
l₃
×
m₃
)
÷
R₀
1.75 = (0 × 1 × 0 + 1 × 0.5 × 1 + 2 × 0.25 × 2 + 3 × 0.1 × 2) ÷ 1.2
Assumptions
•
lₓ is the share of newborn females still alive at age x (years), so l₀ = 1 and lₓ never rises.
•
mₓ is the fecundity: the average daughters a female has while age x.
•
R₀ > 1 means each female more than replaces herself, so the population grows.
•
The generation time T is the mean age of the mothers, weighted by lₓmₓ.
Step-by-step
WE KNOW
l₀ = 1, l₁ = 0.5, l₂ = 0.25, l₃ = 0.1, m₀ = 0, m₁ = 1, m₂ = 2, m₃ = 2
FIND
net reproductive rate (R₀), generation time (T)
Step 1 · Find net reproductive rate (R₀)
Use R₀ = l₀ × m₀ + l₁ × m₁ + l₂ × m₂ + l₃ × m₃
Multiply survival by fecundity at each age, then add the ages.
R₀
=
1
×
0
+
0.5
×
1
+
0.25
×
2
+
0.1
×
2
R₀
=
0.5
+
0.5
+
0.2
R₀
=
1.2
Step 2 · Find generation time (T)
Use T = (0 × l₀ × m₀ + 1 × l₁ × m₁ + 2 × l₂ × m₂ + 3 × l₃ × m₃) ÷ R₀
Weight
each
age
by
its
lₓmₓ,
add,
then
divide
by
R₀
.
T
=
(0
×
1
×
0
+
1
×
0.5
×
1
+
2
×
0.25
×
2
+
3
×
0.1
×
2)
÷
1.2
T
=
0 + 0.5 + 1 + 0.6
1.2
T
=
2.1
1.2
T
=
1.75
yr
Check
Put the numbers back in. Both sides should match.
1.2 = 1 × 0 + 0.5 × 1 + 0.25 × 2 + 0.1 × 2 ✓
1.75 = (0 × 1 × 0 + 1 × 0.5 × 1 + 2 × 0.25 × 2 + 3 × 0.1 × 2) ÷ 1.2 ✓
Topic check
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