Biology · Introductory biology · Worked example
Plot a survivorship curve on a log scale
A cohort of 10,000 birds has 7,000, 4,900, 3,430 and 2,401 alive after 1, 2, 3 and 4 years. Show that the death rate is constant, and describe the curve on ordinary and logarithmic axes.
Find each year’s death rate
Divide each year’s deaths by the number alive at its start.
Plot on an ordinary axis
The yearly drops, 3,000, 2,100, 1,470 and 1,029, shrink as the cohort shrinks, so the curve bends like early loss even though the death rate never changes.
Plot on a log scale
Each year multiplies the survivors by 0.7, so log₁₀ nₓ falls by the same amount, log₁₀ 0.7 ≈ −0.155, every year. Equal steps make a straight line.
| Age (years) | nₓ | log₁₀ nₓ |
|---|---|---|
| 0 | 10000 | 4 |
| 1 | 7000 | 3.845 |
| 2 | 4900 | 3.690 |
| 3 | 3430 | 3.535 |
| 4 | 2401 | 3.380 |
Name the curve
A constant death rate at every age is the Type II pattern, and a straight line on a log scale is how to recognize it. An exponential model fits the counts exactly: N = 10,000e^(−0.357t), with rate ln 0.7 ≈ −0.357 per year.
Result
The death rate is 0.3 every year, so the survivors fall by the same factor each year: a straight line on a log scale, the Type II pattern, even though the ordinary plot curves.
Your turn
A cohort loses 10% of its survivors every year: 1,000, 900, 810, 729. By how much does log₁₀ nₓ fall each year?
Show the answer and explanation
By about 0.046 each year, so the curve is a straight line on a log scale.
Each year multiplies the count by 0.9, and log₁₀ 0.9 ≈ −0.0458. For example, log₁₀ 1,000 = 3 and log₁₀ 900 ≈ 2.954.
Keep exploring
The survivorship tool opens with this cohort, shows qₓ = 0.3 in every year and describes the curve as Type II. In Statistics, an exponential fit to the counts gives the rate −0.356675 per year with r² = 1.
Return to the concept →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
Try in the workspace
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