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Equation 6 · From Origins to Frontier: A History of Evolutionary Biology and Ecology

What does this equation mean?

pt+1=pt(1+s)1+s pt.p_{t+1} = \frac{p_t(1+s)}{1 + s\,p_t}.

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Start withp_t(1+s)
Divide by1 + sp_t
This relates top_t+1
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.

Read it piece by piece

pt+1p_{t+1}

Symbol p_t+1

one classical discrete-generation recursion.

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ptp_t

Symbol p_t

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

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ss

Symbol s

the constant.

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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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addition

addition

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

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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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pt(1+s)p_t(1+s)

Numerator: p_t(1+s)

The complete quantity above the fraction bar.

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1+s pt1 + s\,p_t

Denominator: 1 + sp_t

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

A basic case illustrates the kind of prediction this apparatus makes possible. For a single locus with two alleles under simple directional selection with selection coefficient s favoring an allele at frequency p , one classical discrete-generation recursion is pt+1=pt(1+s)1+s ptp_{t+1} = \frac{p_t(1+s)}{1 + s\,p_t}. This is a deliberately simplified model — haploid, one locus, constant s , no drift, no linkage — and real populations violate several of its assumptions routinely. Its value is not as a literal forecast of any specific population but as a baseline against which departures (frequency-dependent selection, epistasis, drift-dominated dynamics in small populations) can be identified and measured.

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Sources cited in the article section

These citations give research context. Read each source to check which claims it supports.

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