---
title: "Speciation and Macroevolution"
book: "University Biology — Year 2"
subject: biology
language: en
chapter: 23
exercises: 12
source: https://one-course.com/books/biology/4/en/chapter/23-speciation-and-macroevolution
---

# Chapter 23 — Speciation and Macroevolution

On a small island in the Galápagos, in 1982, a few large-beaked ground finches arrived from a neighbouring island and stayed. For twenty years they competed with the resident medium ground finch for the large [seeds](https://one-course.com/books/biology/4/en/chapter/6-sexual-reproduction-of-flowering-plants#def-b2-angiosperm-reproduction-seed) both preferred; then a drought struck, the large [seeds](https://one-course.com/books/biology/4/en/chapter/6-sexual-reproduction-of-flowering-plants#def-b2-angiosperm-reproduction-seed) ran short, and the residents with the larger beaks — those that competed most directly with the newcomers — starved. Within one generation the residents’ beaks had shrunk, measurably, away from the invaders’. Two species that had met were pushed apart. This chapter is about the making of species: what a species is and what keeps it distinct, how one population becomes two, the competition that drives populations apart when they meet, and the long view of the fossil record, where the same process repeated over a hundred million times gives the tree of life.

## 23.1 What a species is

**Definition 23.1 (Species).**

Under the *biological species concept*, a species is a group of populations whose members can interbreed with one another and produce fertile offspring, and are *reproductively isolated* from other such groups: what unites a species is [gene flow](https://one-course.com/books/biology/4/en/chapter/22-population-genetics-and-natural-selection#prop-b2-population-genetics-migration), and what separates species is its absence. The concept works for the sexual organisms of one time and place, which is most of what a biologist meets; it fails for asexual lineages (which are named by resemblance), for fossils (which cannot be crossed), and for the many cases where the boundary is leaky — oaks and willows that hybridise across species, ducks whose species interbreed in captivity, ring species in which neighbouring populations interbreed all the way round a barrier until the ends meet and do not. Other concepts fill the gaps: the *morphological* species (a cluster of like forms), the *phylogenetic* (the smallest diagnosable lineage), the *ecological* (a lineage occupying its own niche). None is wrong; each measures something real, and speciation is the process by which they come to agree.

**Proposition 23.2 (Isolating mechanisms).**

[Gene flow](https://one-course.com/books/biology/4/en/chapter/22-population-genetics-and-natural-selection#prop-b2-population-genetics-migration) between two populations is stopped by barriers acting before or after fertilisation. *Prezygotic*: the populations live in different habitats or breed at different seasons (temporal isolation: two frogs calling in different months); they do not recognise each other’s courtship (behavioural: the songs of crickets, the light codes of fireflies, the dances of ducks); their genitalia or [flowers](https://one-course.com/books/biology/4/en/chapter/6-sexual-reproduction-of-flowering-plants#def-b2-angiosperm-reproduction-flower) do not fit (mechanical); their gametes do not fuse (gametic: the sperm-binding proteins of sea urchins, the $S$ [alleles](https://one-course.com/books/biology/4/en/chapter/4-meiosis-genetic-mixing-and-heredity#def-b2-meiosis-heredity-vocabulary) of [Chapter 6](https://one-course.com/books/biology/4/en/chapter/6-sexual-reproduction-of-flowering-plants#ch-b2-angiosperm-reproduction)). *Postzygotic*: the hybrid dies as an embryo, or lives but is sterile (the mule; the hybrids of two *Drosophila* species), or is fertile but its own offspring are not (hybrid breakdown). *Haldane’s rule*: when only one sex of a hybrid is sterile or inviable, it is the sex with two different [sex chromosomes](https://one-course.com/books/biology/4/en/chapter/4-meiosis-genetic-mixing-and-heredity#prop-b2-meiosis-heredity-sexlinked) — the male in mammals and flies, the female in birds and butterflies — because [recessive](https://one-course.com/books/biology/4/en/chapter/4-meiosis-genetic-mixing-and-heredity#def-b2-meiosis-heredity-vocabulary) incompatibilities on the X are exposed in it. Prezygotic barriers are cheaper, since no gametes are wasted, and selection builds them where the postzygotic ones already exist.

**Theorem 23.3 (Why hybrids fail: the two-locus model).**

Two populations start with [genotype](https://one-course.com/books/biology/4/en/chapter/4-meiosis-genetic-mixing-and-heredity#def-b2-meiosis-heredity-vocabulary) $aabb$. In one, $a$ is replaced by $A$; in the other, $b$ by $B$; each substitution is neutral or favourable in its own background, so neither population ever passes through a less fit state. But $A$ and $B$ have never been tested together, and if the combination is incompatible — if the protein $A$ makes cannot work with the protein $B$ makes — the hybrid $AaBb$ is unfit. [Reproductive isolation](#def-b2-speciation-species) thus arises as a by-product of divergence, with no valley crossed and no selection for isolation itself; and because every new substitution in one lineage can be incompatible with every substitution already fixed in the other, the number of possible incompatibilities grows as the *square* of the number of substitutions — isolation *snowballs* with time.

**Proof.** With $n$ substitutions in each lineage, each of the $n$ new [alleles](https://one-course.com/books/biology/4/en/chapter/4-meiosis-genetic-mixing-and-heredity#def-b2-meiosis-heredity-vocabulary) of one lineage may be incompatible with each of the $n$ of the other: $n^{2}$ pairs, each incompatible with some probability $\varepsilon$, so the expected number of incompatibilities is $\varepsilon n^{2}$, and $n$ grows with the time since the split. The hybrid is the only [genotype](https://one-course.com/books/biology/4/en/chapter/4-meiosis-genetic-mixing-and-heredity#def-b2-meiosis-heredity-vocabulary) in which the two sets meet. ∎

**Evidence.** Coyne and Orr (1989, 1997) compiled hundreds of pairs of *Drosophila* species with known genetic distance (a measure of time since divergence) and measured their prezygotic and [postzygotic isolation](#prop-b2-speciation-isolation): both rise with distance, isolation is essentially complete at a distance corresponding to a few million years, and pairs that live in the same region show stronger [prezygotic isolation](#prop-b2-speciation-isolation) at a given distance than pairs that do not — the signature of reinforcement. Incompatibility genes have since been identified one by one, and the number found between two species does grow faster than linearly with their divergence. ∎

![The two-locus model of hybrid failure. Each population fixes a change that works in its own background; the hybrid is the first genotype to combine the two.](https://one-course.com/images/onecourse/chapters/biology-4/b2-speciation/fig-dd04a82d1910.svg)

*The two-locus model of hybrid failure. Each population fixes a change that works in its own background; the hybrid is the first [genotype](https://one-course.com/books/biology/4/en/chapter/4-meiosis-genetic-mixing-and-heredity#def-b2-meiosis-heredity-vocabulary) to combine the two.*

## 23.2 How one species becomes two

**Proposition 23.4 (Modes of speciation).**

*Allopatric* speciation begins with a geographical barrier: a range split by a rising mountain, a drying sea or a new river (*vicariance*), or a few colonists carried to an island (*peripatric*, with the [founder effect](#prop-b2-speciation-modes) and drift of [Chapter 22](https://one-course.com/books/biology/4/en/chapter/22-population-genetics-and-natural-selection#ch-b2-population-genetics) thrown in). Cut off from [gene flow](https://one-course.com/books/biology/4/en/chapter/22-population-genetics-and-natural-selection#prop-b2-population-genetics-migration), the two populations diverge by selection in different environments, by drift and by their separate [mutations](https://one-course.com/books/biology/4/en/chapter/3-mutations-and-genome-diversification#def-b2-genome-diversification-mutation), until incompatibilities have accumulated; if they meet again they no longer merge. It is the commonest mode, and the pattern of sister species on opposite sides of every isthmus, mountain range and strait is its record. *Sympatric* speciation happens without a barrier: at one step by [polyploidy](https://one-course.com/books/biology/4/en/chapter/3-mutations-and-genome-diversification#def-b2-genome-diversification-polyploidy) ([Chapter 3](https://one-course.com/books/biology/4/en/chapter/3-mutations-and-genome-diversification#ch-b2-genome-diversification)) — a third of plant species arose this way; or by disruptive selection with assortative mating, when a population exploits two resources and individuals mate on the resource they use (the apple maggot fly that shifted from hawthorn to apple in the 1860s and now mates on its host; the cichlids of a crater lake). *Parapatric* speciation occurs along an environmental gradient, where a population adapts to the two ends and a [hybrid zone](#prop-b2-speciation-reinforcement) forms in the middle. In every mode the recipe is the same: reduce [gene flow](https://one-course.com/books/biology/4/en/chapter/22-population-genetics-and-natural-selection#prop-b2-population-genetics-migration), let divergence accumulate, and let the by-products of divergence become barriers.

![Three roads to two species: a barrier through the range, a founder population beyond it, or two resources within it.](https://one-course.com/images/onecourse/chapters/biology-4/b2-speciation/fig-0cf86da097a5.svg)

*Three roads to two species: a barrier through the range, a founder population beyond it, or two resources within it.*

**Proposition 23.5 (Reinforcement and hybrid zones).**

When two diverged populations meet again, three things can happen. If their hybrids are as fit as the parents, they merge. If the hybrids are unfit, selection favours individuals that refuse to mate with the other kind — the wasted gametes are the cost — and [prezygotic isolation](#prop-b2-speciation-isolation) strengthens where the two overlap (*reinforcement*), until they are species. Between these, a *[hybrid zone](#prop-b2-speciation-reinforcement)* persists: a band, often a few kilometres wide, where the two forms interbreed and hybrids are produced every generation, maintained by the balance between dispersal into the zone and selection against the hybrids within it. Across such a zone the frequency of each form follows a *cline*, an S-shaped curve whose width $w$ is set by the dispersal distance $\sigma$ per generation and the selection $s$ against hybrids, $w \approx
\sigma\sqrt{8/s}$: a zone can stay narrow for thousands of years without the two forms ever fusing or fully separating. The toads, crows, mice and grasshoppers of Europe show dozens of them, most running along the lines where populations that had sheltered in separate refuges during the last ice age met again as the ice withdrew.

**Evidence.** Where the pied and collared flycatchers overlap in central Europe, the males of each species have plumage more different from the other’s than where they live apart, and females discriminate more sharply — reinforcement measured in the field. Fire-bellied and yellow-bellied toads meet along a zone across Poland a few kilometres wide, stable for as long as it has been studied, with the cline width predicted from their dispersal and the reduced [fitness](https://one-course.com/books/biology/4/en/chapter/22-population-genetics-and-natural-selection#def-b2-population-genetics-fitness) of their hybrids. ∎

![A cline across a hybrid zone: the frequency of one form falls from one to zero over a width set by how far individuals disperse and how unfit their hybrids are.](https://one-course.com/images/onecourse/chapters/biology-4/b2-speciation/fig-9c017e176529.svg)

*A cline across a [hybrid zone](#prop-b2-speciation-reinforcement): the frequency of one form falls from one to zero over a width set by how far individuals disperse and how unfit their hybrids are.*

## 23.3 Competition, coexistence and character displacement

**Theorem 23.6 (Lotka–Volterra competition).**

Two species with logistic growth (the Year 1 volume) that compete for a resource obey

$$
\frac{\mathrm{d}N_1}{\mathrm{d}t} = r_1 N_1\Bigl(1 - \frac{N_1 + \alpha N_2}{K_1}\Bigr), \qquad
\frac{\mathrm{d}N_2}{\mathrm{d}t} = r_2 N_2\Bigl(1 - \frac{N_2 + \beta N_1}{K_2}\Bigr),
$$

where $\alpha$ measures the effect of one individual of species 2 on species 1 in units of species 1’s own individuals, and $\beta$ the reverse. Species 1 stops growing on the line $N_1 + \alpha N_2 =
K_1$ (its *isocline*), species 2 on $N_2 + \beta N_1 = K_2$, and the outcome depends on how the two lines cross:

- $K_1 > \alpha K_2$ and $K_2 < \beta K_1$ : species 1 wins always;
- $K_1 < \alpha K_2$ and $K_2 > \beta K_1$ : species 2 wins always;
- $K_1 > \alpha K_2$ and $K_2 > \beta K_1$ : *stable coexistence* at $N_1^{*} = (K_1 - \alpha K_2)/(1 - \alpha\beta)$ , $N_2^{*} = (K_2 - \beta K_1)/(1 - \alpha\beta)$ ;
- $K_1 < \alpha K_2$ and $K_2 < \beta K_1$ : either wins, depending on who starts more numerous.

Coexistence requires $\alpha\beta < 1$ with each species held back more by its own kind than by the other: two species that use exactly the same resource ($\alpha = \beta = 1$) cannot coexist (the *[competitive exclusion](#thm-b2-speciation-lv) principle*), and the more they differ in what they use, the smaller $\alpha$ and $\beta$ and the more securely they coexist.

**Proof.** On species 1’s isocline $\mathrm{d}N_1/\mathrm{d}t = 0$; below it (toward the origin) $N_1$ grows, above it $N_1$ falls; likewise for species 2. Each isocline is a straight line with intercepts $K_1$ on the $N_1$ axis and $K_1/\alpha$ on the $N_2$ axis (species 1’s), $K_2/\beta$ and $K_2$ (species 2’s). Species 1 can invade a population of species 2 at its carrying capacity $K_2$ if its own growth is positive there, i.e. if $K_1 > \alpha K_2$; species 2 can invade species 1 at $K_1$ if $K_2 > \beta K_1$. When both can invade the other, neither can be excluded and the trajectories converge to the crossing of the isoclines, found by solving the two line equations; when only one can, it wins; when neither can, the crossing is a saddle and the founder wins. Stability of the interior point follows from the invasion argument: perturbed toward either axis, the rarer species grows back. ∎

![The phase plane of two competitors with K_1 = 1000, K_2 = 800, = 0.5, = 0.6: each species can invade the other’s carrying capacity, and the trajectories converge on the crossing of the isoclines at N_1* = 857, N_2* = 286.](https://one-course.com/images/onecourse/chapters/biology-4/b2-speciation/fig-a05dee0b56ea.svg)

*The phase plane of two competitors with $K_1 = 1000$, $K_2 =
800$, $\alpha = 0.5$, $\beta = 0.6$: each species can invade the other’s carrying capacity, and the trajectories converge on the crossing of the isoclines at $N_1^{*} = 857$, $N_2^{*} = 286$.*

**Proposition 23.7 (Character displacement).**

Competition is not only an ecological outcome but an evolutionary force. Where two similar species meet, the individuals of each that most resemble the other suffer most from the competition, and selection pushes the two apart in the character that governs the resource use — beak size, body size, flowering time: the populations that overlap differ more than the populations that live alone. This *[character displacement](#prop-b2-speciation-displacement)* shrinks $\alpha$ and $\beta$, converts a contest for exclusion into coexistence, and, repeated across a landscape of opportunities, produces an *[adaptive radiation](#prop-b2-speciation-displacement)*: one colonist becoming a dozen species each fitted to a resource — Darwin’s finches on the Galápagos, the honeycreepers of Hawaii, the cichlids of the African lakes (five hundred species in Lake Victoria in under fifteen thousand years), the anole lizards of the Caribbean, where the same set of niches has been filled by the same set of body forms independently on four islands.

**Evidence.** The Grants recorded every finch on Daphne Major for forty years. After the large ground finch established itself in 1982, the medium ground finch’s beak did not change until the drought of 2004, when large [seeds](https://one-course.com/books/biology/4/en/chapter/6-sexual-reproduction-of-flowering-plants#def-b2-angiosperm-reproduction-seed) became scarce and the two species competed for them; the medium finches with the largest beaks died in disproportion, and the next generation’s beaks were smaller by more than had ever been seen in a single step — the [character displacement](#prop-b2-speciation-displacement) predicted by theory, observed as it happened, with its selection differential and heritability measured. The same finches’ beaks, compared between islands where they live alone and islands where they meet, show the same displacement written across the archipelago. ∎

![Two radiations. Left: Darwin’s finches as drawn by Gould in 1845, one colonist become many, distinguished by their beaks. Right: cichlids of an African lake, hundreds of species from a few founders, each with its own mouth and diet.](https://one-course.com/images/onecourse/chapters/biology-4/b2-speciation/img-bd51f01356a3.jpg)

![Two radiations. Left: Darwin’s finches as drawn by Gould in 1845, one colonist become many, distinguished by their beaks. Right: cichlids of an African lake, hundreds of species from a few founders, each with its own mouth and diet.](https://one-course.com/images/onecourse/chapters/biology-4/b2-speciation/img-561d286cfdbf.jpg)

*Two radiations. Left: Darwin’s finches as drawn by Gould in 1845, one colonist become many, distinguished by their beaks. Right: cichlids of an African lake, hundreds of species from a few founders, each with its own mouth and diet.*

## 23.4 Macroevolution: the long view

**Proposition 23.8 (Tempo and mode).**

Over a few dozen generations the moth changed colour and the finch its beak; over millions of years the fossil record shows lineages that hardly change for long stretches and then, in a geological instant of a few thousand years, are replaced by a distinct descendant — *[punctuated equilibrium](#prop-b2-speciation-tempo)*, which is what [allopatric speciation](#prop-b2-speciation-modes) in small peripheral populations followed by their spread looks like when sampled every hundred thousand years; and it shows other lineages, the horses among them, that change gradually across dozens of species. Above the species, the record is one of turnover: most species last one to ten million years, the *background extinction* rate is about one species in a million per year, and five times — at the end of the Ordovician, Devonian, Permian (when nine tenths of marine species vanished), Triassic and Cretaceous — a *[mass extinction](#prop-b2-speciation-tempo)* removed most of what lived within a few hundred thousand years and reset the game, the survivors radiating into the emptied niches: the mammals into the dinosaurs’ world after the asteroid of 66 million years ago. Macroevolution is the pattern that the microevolution of this and the previous chapter, run for a very long time and interrupted by catastrophe, has drawn.

**Example 23.9 (Rates compared).**

Selection on the finches changed beak depth by $4\,\%$ in one generation; the whole Galápagos radiation, fourteen species from one founder, took two million years; the difference between a mouse and an elephant in body size, some $10^{5}$, could be produced by selection of the finch strength in a few hundred generations — and the record shows it took forty million years. The paradox is resolved by noticing that selection rarely pushes the same way for long: it reverses with the weather, as it did on Daphne Major the year after the drought, and [stabilising selection](https://one-course.com/books/biology/4/en/chapter/22-population-genetics-and-natural-selection#prop-b2-population-genetics-kinds) holds most lineages where they are. Sustained change is rare, which is why speciation, which needs it, is slow — and why, when the world changes fast, as it does in [Chapter 27](https://one-course.com/books/biology/4/en/chapter/27-global-change-and-the-biosphere#ch-b2-global-change), the lineages that cannot follow are lost.

## 23.5 Exercises

**Exercise 23.1 ★.**

State the [biological species concept](#def-b2-speciation-species) and give three cases in which it cannot be applied, with the concept used instead.

**Solution of Exercise 23.1.**

A species is a group of interbreeding natural populations reproductively isolated from other such groups. It fails for asexual lineages (bacteria, bdelloid rotifers, dandelions: morphological or phylogenetic concept, clusters of like forms or diagnosable lineages); for fossils (morphological concept, since no cross can be made); and for allopatric populations that never meet (ecological or phylogenetic concept, since interbreeding cannot be tested) — and it is blurred by hybridising oaks and by ring species.

**Exercise 23.2 ★.**

Classify as prezygotic or postzygotic: two frogs breeding in different months; a mule; a sea urchin sperm that cannot bind another species’ egg; hybrid plants that are vigorous but whose [seeds](https://one-course.com/books/biology/4/en/chapter/6-sexual-reproduction-of-flowering-plants#def-b2-angiosperm-reproduction-seed) do not germinate; two crickets with different songs.

**Solution of Exercise 23.2.**

Frogs in different months: prezygotic (temporal). Mule: postzygotic (hybrid sterility). Sperm that cannot bind: prezygotic (gametic). Vigorous hybrids with sterile [seed](https://one-course.com/books/biology/4/en/chapter/6-sexual-reproduction-of-flowering-plants#def-b2-angiosperm-reproduction-seed): postzygotic (hybrid sterility, or hybrid breakdown if the failure is in the next generation). Crickets with different songs: prezygotic (behavioural).

**Exercise 23.3 ★.**

Define allopatric, peripatric, sympatric and [parapatric speciation](#prop-b2-speciation-modes), with an example of each.

**Solution of Exercise 23.3.**

Allopatric: a barrier divides a range (snapping shrimps on the two sides of the Isthmus of Panama). Peripatric: a few founders beyond the range (Darwin’s finches from a mainland ancestor). Sympatric: no barrier (the apple maggot fly on hawthorn and apple; an [allopolyploid](https://one-course.com/books/biology/4/en/chapter/3-mutations-and-genome-diversification#def-b2-genome-diversification-polyploidy) wheat). Parapatric: along a gradient with a [hybrid zone](#prop-b2-speciation-reinforcement) in the middle (grasses on and off mine tailings, flowering at different times).

**Exercise 23.4 ★.**

State the four outcomes of [Lotka–Volterra competition](#thm-b2-speciation-lv) and the condition for stable coexistence in words.

**Solution of Exercise 23.4.**

Species 1 always wins; species 2 always wins; stable coexistence; either wins according to initial numbers. Coexistence is stable when each species can invade the other at its carrying capacity, that is when each species limits itself more than it limits the other ($K_1 > \alpha K_2$ and $K_2 > \beta K_1$).

**Exercise 23.5 ★★.**

With $K_1 = 1000$, $K_2 = 800$, $\alpha = 0.5$, $\beta = 0.6$, compute the equilibrium densities and check that each species can invade the other’s carrying capacity.

**Solution of Exercise 23.5.**

$\alpha K_2 = 400 < K_1 = 1000$: species 1 invades; $\beta K_1 = 600
< K_2 = 800$: species 2 invades. $N_1^{*} = (1000 - 400)/(1 - 0.3) =
857$, $N_2^{*} = (800 - 600)/0.7 = 286$; at the equilibrium species 1 is below its $K$ by $\alpha N_2^{*} = 143$ and species 2 by $\beta
N_1^{*} = 514$.

**Exercise 23.6 ★★.**

With $K_1 = 1000$, $K_2 = 800$, $\alpha = 1.5$, $\beta = 0.3$, which species wins, and why? What if $\alpha = 1.5$ and $\beta = 1.5$?

**Solution of Exercise 23.6.**

$\alpha K_2 = 1200 > K_1$: species 1 cannot invade species 2 at carrying capacity; $\beta K_1 = 300 < K_2$: species 2 can invade species 1. Species 2 wins always, because its individuals hurt species 1 more than species 1’s own do while it is itself barely affected. With $\alpha = \beta = 1.5$: $\alpha K_2 = 1200 > 1000$ and $\beta K_1 = 1500 > 800$: neither can invade the other, each excludes the other when established, and the founder wins.

**Exercise 23.7 ★★.**

A tetraploid arises in a [diploid](https://one-course.com/books/biology/4/en/chapter/4-meiosis-genetic-mixing-and-heredity#def-b2-meiosis-heredity-meiosis) population. Give the chromosome numbers of its gametes, of a cross with the [diploid](https://one-course.com/books/biology/4/en/chapter/4-meiosis-genetic-mixing-and-heredity#def-b2-meiosis-heredity-meiosis), and of that hybrid’s gametes, and explain why the tetraploid is isolated at once. Why does it nevertheless usually fail to persist?

**Solution of Exercise 23.7.**

Tetraploid $4n$ makes $2n$ gametes; crossed with the [diploid](https://one-course.com/books/biology/4/en/chapter/4-meiosis-genetic-mixing-and-heredity#def-b2-meiosis-heredity-meiosis)’s $n$ gametes it gives triploids $3n$, whose [meiosis](https://one-course.com/books/biology/4/en/chapter/4-meiosis-genetic-mixing-and-heredity#def-b2-meiosis-heredity-meiosis) cannot pair three sets and yields aneuploid, mostly inviable gametes. The tetraploid can breed only with itself (or by selfing): isolated in one step. It usually fails because it is alone — its only potential mates are [diploids](https://one-course.com/books/biology/4/en/chapter/4-meiosis-genetic-mixing-and-heredity#def-b2-meiosis-heredity-meiosis) and every such cross is wasted — and is swamped; it persists when it selfs, propagates vegetatively, or arises repeatedly.

**Exercise 23.8 ★★.**

Two populations have each fixed 10 substitutions since they split, and each pair of new [alleles](https://one-course.com/books/biology/4/en/chapter/4-meiosis-genetic-mixing-and-heredity#def-b2-meiosis-heredity-vocabulary) from the two sides is incompatible with probability $0.01$. Compute the expected number of incompatibilities now, and after 30 substitutions each. What does this predict for the relation of isolation to time?

**Solution of Exercise 23.8.**

$0.01\times 10\times 10 = 1$ incompatibility; with 30 each, $0.01\times
900 = 9$. Isolation grows as the square of divergence time: two populations diverged for three times as long are nine times as isolated — the snowball.

**Exercise 23.9 ★★.**

Compute the width of a [hybrid zone](#prop-b2-speciation-reinforcement) for dispersal $\sigma =
2\,\mathrm{km}$ per generation and hybrid disadvantage $s = 0.05$, and for $\sigma = 0.3\,\mathrm{km}$ and $s = 0.4$. Which zone is more likely to end in reinforcement, and why?

**Solution of Exercise 23.9.**

$w = 2\sqrt{8/0.05} = 25\,\mathrm{km}$; $w = 0.3\sqrt{8/0.4} =
1.3\,\mathrm{km}$. Reinforcement is likelier in the second: hybrids are strongly unfit, so a female that refuses the other kind gains much, and the zone is narrow enough for the two forms to meet often enough for that refusal to be selected.

**Exercise 23.10 ★★★.**

Two finch species compete for [seeds](https://one-course.com/books/biology/4/en/chapter/6-sexual-reproduction-of-flowering-plants#def-b2-angiosperm-reproduction-seed) with $\alpha = \beta = 0.9$ and $K_1 = K_2 = 500$. Do they coexist? [Character displacement](#prop-b2-speciation-displacement) reduces $\alpha$ and $\beta$ to $0.4$: recompute the equilibrium. Explain in terms of the isoclines what displacement did.

**Solution of Exercise 23.10.**

$\alpha K_2 = 450 < 500$, so each invades the other: they coexist, at $N^{*} = (500 - 450)/(1 - 0.81) = 263$ each — barely, near the edge of exclusion, and a small change in $K$ would tip it. With $\alpha = \beta = 0.4$: $N^{*} = (500 - 200)/(1 - 0.16) = 357$ each. Displacement rotated each isocline away from the other’s ($K/\alpha$ moved outward), so the crossing moved away from the axes and each species sits nearer its own $K$: a more secure coexistence with more of each.

**Exercise 23.11 ★★★.**

The Grants found that after the 2004 drought the medium ground finch’s mean beak depth fell by $0.7\,\mathrm{mm}$ with $h^{2} = 0.7$, the survivors having been $1\,\mathrm{mm}$ smaller-beaked than the population. Check the response with the [breeder’s equation](https://one-course.com/books/biology/4/en/chapter/22-population-genetics-and-natural-selection#thm-b2-population-genetics-heritability), and explain why the large ground finch’s presence, not the drought alone, caused the direction of the change.

**Solution of Exercise 23.11.**

$R = h^{2}S = 0.7\times(-1) = -0.7\,\mathrm{mm}$: as observed. In the 1977 drought, with no large ground finch present, the same medium finches were selected for *larger* beaks, since large [seeds](https://one-course.com/books/biology/4/en/chapter/6-sexual-reproduction-of-flowering-plants#def-b2-angiosperm-reproduction-seed) were the food left; in 2004 the large [seeds](https://one-course.com/books/biology/4/en/chapter/6-sexual-reproduction-of-flowering-plants#def-b2-angiosperm-reproduction-seed) were taken by the invader, and the medium finches that competed for them lost: the direction of selection was set by the competitor.

**Exercise 23.12 ★★★.**

“Species are made by accident and kept by selection.” Discuss, with the two-locus model, reinforcement, and the role of geography.

**Solution of Exercise 23.12.**

Made by accident: the two-locus model shows isolation arising as a by-product of substitutions fixed for other reasons, or by drift, and geography — a barrier, a founding — supplies the separation that lets the accident accumulate. Kept by selection: where the forms meet, selection against unfit hybrids builds prezygotic barriers (reinforcement), and competition displaces characters so that the two can coexist. The slogan omits [sympatric speciation](#prop-b2-speciation-modes) by disruptive selection, where selection makes as well as keeps the species, and [polyploidy](https://one-course.com/books/biology/4/en/chapter/3-mutations-and-genome-diversification#def-b2-genome-diversification-polyploidy), where the accident is the whole story.

## 23.6 Problem: Two Finches on an Island

**Problem 23.1.**

Weekend problem — a resident finch meets an invader: their competition analysed on the phase plane, the displacement of the resident’s beak predicted from the breeder’s equation, and the isolation between them measured against a hybrid zone, ending on the equilibrium densities, the beak’s shift, and the cline width

Resident species 1: $K_1 = 1200$, $r_1 = 0.5$ per year; invader species 2: $K_2 = 400$. Before displacement $\alpha = 0.9$, $\beta =
0.3$; a drought halves both carrying capacities for one year. Beak depth of species 1: mean $9.5\,\mathrm{mm}$, standard deviation $0.8\,\mathrm{mm}$, $h^{2} = 0.7$. [Hybrid zone](#prop-b2-speciation-reinforcement) parameters: $\sigma =
0.5\,\mathrm{km}$, $s = 0.2$.

**Part I — The invasion.**

1. Write the two isoclines and their intercepts.
2. Can species 2 invade species 1 at $K_1$ ? Can species 1 invade species 2 at $K_2$ ? Conclude on the outcome.
3. Compute the equilibrium densities.
4. Compute $\mathrm{d}N_1/\mathrm{d}t$ at the moment species 2 arrives with $N_2 = 20$ and $N_1 = 1200$ . What does its sign mean?
5. During the drought both $K$ are halved. Recompute the equilibrium and say what happens to each species’ numbers.
6. Why is the invader’s effect on the resident ( $\alpha = 0.9$ ) larger than the resident’s on the invader ( $\beta = 0.3$ )?

**Part II — Displacement.** In the drought the resident individuals with the largest beaks compete most with the invader and die: the survivors have a mean beak of $9.0\,\mathrm{mm}$.

7. Compute the selection differential and the response.
8. Give the next generation’s mean beak.
9. After the shift, $\alpha$ falls to $0.5$ . Recompute the equilibrium (with the original $K$ ).
10. Explain what the shift did to species 1’s isocline.
11. If selection of this strength acted every year, how many years would it take to shift the beak by $2\,\mathrm{mm}$ ? Why does it not?
12. Compare with the two-million-year age of the radiation and comment.

**Part III — Isolation.** The two species can hybridise; hybrids have intermediate beaks and [fitness](https://one-course.com/books/biology/4/en/chapter/22-population-genetics-and-natural-selection#def-b2-population-genetics-fitness) $1 - s$.

13. Name the prezygotic barrier that keeps hybridisation rare in Darwin’s finches, and one postzygotic cost of the hybrids.
14. Compute the width of a [hybrid zone](#prop-b2-speciation-reinforcement) between them for $\sigma  = 0.5\,\mathrm{km}$ , $s = 0.2$ .
15. The island is $1\,\mathrm{km}$ across. What does that imply?
16. Explain how reinforcement would proceed, and what character it would act on.
17. Two populations of the resident on different islands have each fixed 15 substitutions since they separated; pairs are incompatible with probability $0.005$ . Compute the expected incompatibilities and say whether they are species.
18. Would the [biological species concept](#def-b2-speciation-species) , the morphological concept and the phylogenetic concept agree about the two island populations? Explain.

**Part IV — The long view.**

19. If a species lasts on average 5 million years, what is the background extinction rate per species per year? For a biota of 10 million species, how many extinctions a year?
20. The current extinction rate is estimated at 100 to 1000 times background. How many species a year is that?
21. A [mass extinction](#prop-b2-speciation-tempo) removes $75\,\%$ of species in $100\,000$ years. Compare its rate with the present one.
22. Explain in one sentence why the survivors of a [mass extinction](#prop-b2-speciation-tempo) radiate.
23. A lineage in the record changes little for 3 million years, then is replaced within $20\,000$ years by a distinct descendant. Name the pattern and give the speciation mode that produces it.
24. Using the finch’s $4\,\%$ per generation, how many generations of sustained selection would double beak depth? Why does the fossil record show such changes taking millions of years?
25. State the result: the outcome and equilibrium of the competition before and after displacement, the beak’s shift in one generation, and the [hybrid zone](#prop-b2-speciation-reinforcement) width.

**Solution of Problem 23.1.**

**1.** Species 1: $N_1 + 0.9N_2 = 1200$, intercepts $N_1 = 1200$ and $N_2 = 1333$. Species 2: $N_2 + 0.3N_1 = 400$, intercepts $N_2 =
400$ and $N_1 = 1333$. **2.** $\beta K_1 = 360 < K_2 = 400$: species 2 invades; $\alpha K_2 = 360 < K_1 = 1200$: species 1 invades. Stable coexistence. **3.** $N_1^{*} = (1200 - 360)/(1 - 0.27) = 1151$; $N_2^{*} =
(400 - 360)/0.73 = 55$. **4.** $\mathrm{d}N_1/\mathrm{d}t = 0.5\times 1200\,(1 - 1218/1200)
= -9$ per year: the resident, at its carrying capacity, is pushed into decline by the newcomers. **5.** $K_1 = 600$, $K_2 = 200$: $N_1^{*} = (600 - 180)/0.73 =
575$, $N_2^{*} = (200 - 180)/0.73 = 27$: both halve, and the invader, at 27 birds, is close to extinction — the drought is when competition bites. **6.** The invader is the larger bird with the larger beak: it takes the large [seeds](https://one-course.com/books/biology/4/en/chapter/6-sexual-reproduction-of-flowering-plants#def-b2-angiosperm-reproduction-seed) the resident also uses, and takes them faster, while the resident’s small [seeds](https://one-course.com/books/biology/4/en/chapter/6-sexual-reproduction-of-flowering-plants#def-b2-angiosperm-reproduction-seed) are of little use to it. **7.** $S = 9.0 - 9.5 = -0.5\,\mathrm{mm}$; $R = 0.7\times(-0.5) =
-0.35\,\mathrm{mm}$. **8.** $9.5 - 0.35 = 9.15\,\mathrm{mm}$. **9.** $N_1^{*} = (1200 - 200)/(1 - 0.15) = 1176$; $N_2^{*} =
(400 - 360)/0.85 = 47$. **10.** The $N_2$ intercept of species 1’s isocline rose from $K_1/0.9 = 1333$ to $K_1/0.5 = 2400$: the line rotated away from the invader’s, the resident is now less affected by each invader, and the equilibrium moved toward the resident’s $K$. **11.** $2/0.35 \approx 6$ years. It does not because selection of this kind occurs only in drought years, reverses in wet years when small [seeds](https://one-course.com/books/biology/4/en/chapter/6-sexual-reproduction-of-flowering-plants#def-b2-angiosperm-reproduction-seed) abound (or when the invader is scarce), and because the additive variance would be depleted. **12.** Two million years is about $10^{6}$ generations; a shift of $0.35\,\mathrm{mm}$ per generation sustained for even a thousandth of them would change the beak by hundreds of millimetres. Selection is strong but inconstant; sustained directional change is the rare exception, and net change over geological time is a tiny residue of large opposing episodes. **13.** Song, learned from the father and used by females in mate choice, with beak shape itself as a cue. Hybrids with intermediate beaks handle neither the large nor the small [seeds](https://one-course.com/books/biology/4/en/chapter/6-sexual-reproduction-of-flowering-plants#def-b2-angiosperm-reproduction-seed) best and starve first in droughts. **14.** $w = 0.5\sqrt{8/0.2} = 3.2\,\mathrm{km}$. **15.** The island is narrower than the zone would be: no spatial cline can form; hybridisation, where it occurs, affects the whole population, and only assortative mating keeps the two apart. **16.** Females that mate only with males of their own song and beak leave more grandchildren than those that hybridise; the preference and the cue (song, beak) become more distinct where the two species live together than where they live alone. **17.** $0.005\times 15\times 15 = 1.1$ incompatibilities: on average one, with hybrids probably reduced but not sterile. Not yet species by the biological criterion; on the way. **18.** Biological: not yet (hybrids nearly fit). Morphological: perhaps, if the beaks or plumage have diverged visibly. Phylogenetic: yes, if each population has a fixed diagnostic difference. The concepts disagree exactly during the process of speciation, which is what makes them useful. **19.** $1/(5\times 10^{6}) = 2\times 10^{-7}$ per species per year; for $10^{7}$ species, two extinctions a year. **20.** 200 to 2000 species a year. **21.** $0.75\times 10^{7}/10^{5} = 75$ species a year — lower than the present rate at the low end. The present episode, if sustained, is a [mass extinction](#prop-b2-speciation-tempo) proceeding faster than the big five. **22.** The niches vacated by the losers are empty of competitors, so any survivor variant that can use one is favoured, and the survivors diversify into them. **23.** [Punctuated equilibrium](#prop-b2-speciation-tempo); allopatric (peripatric) speciation in a small isolated population whose descendant then spreads into the main population’s range. **24.** $1.04^{n} = 2$: $n = \ln 2/\ln 1.04 = 18$ generations. The record shows millions of years because selection reverses, [stabilising selection](https://one-course.com/books/biology/4/en/chapter/22-population-genetics-and-natural-selection#prop-b2-population-genetics-kinds) dominates most of the time, and net change is the small difference between large opposed episodes. **25.** Before displacement: stable coexistence at $N_1^{*} =
1151$, $N_2^{*} = 55$; after: 1176 and 47. Beak shift $-0.35\,\mathrm{mm}$ in one generation. [Hybrid zone](#prop-b2-speciation-reinforcement) width $3.2\,\mathrm{km}$.