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Equation 9 · The Peacock Problem: Darwin's Second Theory and Its Hard Tests

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Δtˉ=ht2 βt+B βpΔpˉ=B βt+hp2 βp\begin{aligned} \Delta \bar{t} &= h_t^{2}\,\beta_t + B\,\beta_p \\ \Delta \bar{p} &= B\,\beta_t + h_p^{2}\,\beta_p \end{aligned}

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Inputs and operationsh_t^2beta_t + Bbeta_p Δ barp &= Bbeta_t + h_p^2beta_p endaligned
Result or conditionbeginaligned Δ bart &
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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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Δ\Delta

Symbol Δ

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

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tˉ\bar{t}

Symbol bart

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

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ht2h_t^{2}

Symbol h_t^2

ht2h_t^2 is one of the signed contributions combined to compute the quantity on the left.

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βt\beta_t

Symbol beta_t

the write.

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BB

Symbol B

the when.

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βp\beta_p

Symbol beta_p

betapa_p is one of the signed contributions combined to compute the quantity on the left.

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pˉ\bar{p}

Symbol barp

barp is one of the signed contributions combined to compute the quantity on the left.

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hp2h_p^{2}

Symbol h_p^2

hp2h_p^2 is one of the signed contributions combined to compute 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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addition

addition

Add the term after the plus sign to the term or group before it.

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change

change

Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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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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How to interpret it

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What the article says around this equation

Russell Lande’s 1981 paper gave this verbal logic its standard quantitative-genetic form, modelling the joint evolution of a female preference trait and a male display trait under stabilising natural selection on the display and showed that, “despite stabilizing natural selection on males, various types of mating preferences may create a runaway process in which the outcome of phenotypic evolution depends critically on the genetic variation parameters and initial conditions of a population” [ 4 ] . One compact way to see why sits in the coupling itself. Let tˉ\bar t be the population mean of a male display trait and pˉ\bar p the population mean of the corresponding female preference; write…
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Russell Lande’s 1981 paper gave this verbal logic its standard quantitative-genetic form, modelling the joint evolution of a female preference trait and a male display trait under stabilising natural selection on the display and showed that, “despite stabilizing natural selection on males, various types of mating preferences may create a runaway process in which the outcome of phenotypic evolution depends critically on the genetic variation parameters and initial conditions of a population” [ 4 ] . One compact way to see why sits in the coupling itself. Let tˉ\bar t be the population mean of a male display trait and pˉ\bar p the population mean of the corresponding female preference; write βt\beta_t and βp\beta_p for the direct selection gradients acting on each (natural selection pulling βt\beta_t downward, since the ornament is costly), ht2h_t^2 and hp2h_p^2 for their heritabilities, and B for the genetic covariance that non-random mating builds between the two traits. The standard Lande-Fisher formalisation of one generation’s change is then a coupled pair: Δtˉ=ht2 βt+B βpΔpˉ=B βt+hp2 βp\begin{aligned} \Delta \bar{t} &= h_t^{2}\,\beta_t + B\,\beta_p \\ \Delta \bar{p} &= B\,\beta_t + h_p^{2}\,\beta_p \end{aligned}. Read the first line on its own and the trait looks stable: natural selection ( βt\beta_t ) pulls the ornament back down every generation. But the second term, B\,βp\beta_p , imports a push from the preference side through the genetic covariance the mating system itself has created. When B is large enough relative to the stabilising pull of βt\beta_t , the two equations no longer settle to a fixed point; they feed each other, and tˉ\bar t and pˉ\bar p escalate together until some external limit — a wing that can no longer lift the tail, a display that finally costs more than it wins — brings the pair to a halt. Nothing in the equations names where that limit falls. That absence of an internal stopping rule is the entire content of “runaway”: once the covariance term dominates, the direction is set and the magnitude is set by biomechanics, not by the selective logic that started the process.

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