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Equation 1 · The Longest Childhood: Life-History Theory and the Grandmother Bet

What does this equation mean?

dwdt=Aw0.75\frac{dw}{dt} = A w^{0.75}

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Inputs and operationsA w^0.75
Result or conditionfracdwdt
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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dd

Symbol d

d is part of the quantity the equation computes from the expression on the right.

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ww

Symbol w

body weight.

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tt

Symbol t

t occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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AA

Symbol A

a taxon-specific growth constant.

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w0.75w^{0.75}

Symbol w^0.75

w0w^0.75 is an input to the expression that computes the quantity on the left.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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fraction

fraction

Divide the expression above the line by the one below it.

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superscript

superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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dwdw

Numerator: dw

The complete quantity above the fraction bar.

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dtdt

Denominator: dt

The complete quantity below the fraction bar; it must be nonzero for this division.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

Put the three numbers next to each other and the puzzle sharpens rather than resolving. A species that weans early is, on the usual mammalian logic, moving resources toward producing more offspring faster. A species that delays first reproduction by five years relative to a close comparator is, on the same logic, moving resources the other way, into growth and skill before any output. A species in which a large share of the reproductively spent population survives for decades afterward is carrying dependents — grown or not — who cannot be explained by ordinary offspring investment, because offspring investment stops paying by the time the investor can no longer reproduce. The three facts do…
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Put the three numbers next to each other and the puzzle sharpens rather than resolving. A species that weans early is, on the usual mammalian logic, moving resources toward producing more offspring faster. A species that delays first reproduction by five years relative to a close comparator is, on the same logic, moving resources the other way, into growth and skill before any output. A species in which a large share of the reproductively spent population survives for decades afterward is carrying dependents — grown or not — who cannot be explained by ordinary offspring investment, because offspring investment stops paying by the time the investor can no longer reproduce. The three facts do not obviously belong to a single motive. One classic model puts a number on how much of the delay in maturity is simply growth: the allometric relationship between body weight and growth rate that Kaplan and colleagues cite from the wider mammalian literature is dwdt=Aw0.75\frac{dw}{dt} = A w^{0.75}. where w is body weight and A is a taxon-specific growth constant; most mammals run at a constant of about 1, but “the mean primate value for A is about 0.4” [ 3 ] . Primates grow slowly relative to other mammals of the same size to begin with, and humans, on the same equation, grow slower still during middle childhood than even that primate baseline would predict, while gaining weight faster than juvenile chimpanzees do at ages 5 to 10 in absolute terms — 2.6 kilograms a year against 1.6 for chimpanzees over the first five years, reversing to 2.1 against 2.5 kilograms a year for chimpanzees between ages 5 and 10 [ 3 ] . Slow, uneven growth by primate standards, early weaning, late first birth, and a long tail of survivors who are done reproducing: that is the package the rest of this article is trying to explain, and it is expensive enough, in strict energetic terms, that “somebody has to be paying for it” is not a metaphor. It is the actual argument on offer three different ways.

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Sources cited in the surrounding passage

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